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ch6 Interest Rates (solutions_nss_nc_6)

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ch6 Interest Rates (solutions_nss_nc_6)

    Chapter 6

    Interest Rates

    Learning Objectives

After reading this chapter, students should be able to:

    ; Explain how capital is allocated in a supply/demand framework, and list the fundamental factors that

    affect the cost of money.

    ; Write out two equations for the nominal, or quoted, interest rate, and briefly discuss each component. ; Define what is meant by the term structure of interest rates, and graph a yield curve for a given set of

    data.

    ; Explain what factors determine the shape of the yield curve.

    ; Use the yield curve and the information embedded in it to estimate the market’s expectations

    regarding future inflation and risk.

    ; List four additional factors that influence the level of interest rates and the slope of the yield curve. ; Discuss country risk.

    ; Briefly explain how interest rate levels affect business decisions.

Chapter 6: Interest Rates Learning Objectives 117

    Lecture Suggestions

    Chapter 6 is important because it lays the groundwork for the following chapters. Additionally, students have a curiosity about interest rates, so this chapter stimulates their interest in the course.

    What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 6, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the “Lecture Suggestions” in Chapter 2, where we describe how we conduct our classes.

DAYS ON CHAPTER: 1 OF 58 DAYS (50-minute periods)

118 Lecture Suggestions Chapter 6: Interest Rates

    Answers to End-of-Chapter Questions

    6-1 Regional mortgage rate differentials do exist, depending on supply/demand conditions in the

    different regions. However, relatively high rates in one region would attract capital from other

    regions, and the end result would be a differential that was just sufficient to cover the costs of

    effecting the transfer (perhaps ? of one percentage point). Differentials are more likely in the

    residential mortgage market than the business loan market, and not at all likely for the large,

    nationwide firms, which will do their borrowing in the lowest-cost money centers and thereby

    quickly equalize rates for large corporate loans. Interest rates are more competitive, making it

    easier for small borrowers, and borrowers in rural areas, to obtain lower cost loans.

    6-2 Short-term interest rates are more volatile because (1) the Fed operates mainly in the short-term

    sector, hence Federal Reserve intervention has its major effect here, and (2) long-term interest

    rates reflect the average expected inflation rate over the next 20 to 30 years, and this average

    does not change as radically as year-to-year expectations.

    6-3 Interest rates will fall as the recession takes hold because (1) business borrowings will decrease

    and (2) the Fed will increase the money supply to stimulate the economy. Thus, it would be

    better to borrow short-term now, and then to convert to long-term when rates have reached a

    cyclical low. Note, though, that this answer requires interest rate forecasting, which is extremely

    difficult to do with better than 50% accuracy.

    6-4 a. If transfers between the two markets are costly, interest rates would be different in the two

    areas. Area Y, with the relatively young population, would have less in savings accumulation

    and stronger loan demand. Area O, with the relatively old population, would have more

    savings accumulation and weaker loan demand as the members of the older population have

    already purchased their houses and are less consumption oriented. Thus, supply/demand

    equilibrium would be at a higher rate of interest in Area Y.

    b. Yes. Nationwide branching, and so forth, would reduce the cost of financial transfers between

    the areas. Thus, funds would flow from Area O with excess relative supply to Area Y with

    excess relative demand. This flow would increase the interest rate in Area O and decrease

    the interest rate in Y until the rates were roughly equal, the difference being the transfer cost.

    6-5 A significant increase in productivity would raise the rate of return on producers’ investment, thus

    causing the investment curve (see Figure 6-1 in the textbook) to shift to the right. This would

    increase the amount of savings and investment in the economy, thus causing all interest rates to

    rise.

    6-6 a. The immediate effect on the yield curve would be to lower interest rates in the short-term end

    of the market, since the Fed deals primarily in that market segment. However, people would

    expect higher future inflation, which would raise long-term rates. The result would be a much

    steeper yield curve.

    b. If the policy is maintained, the expanded money supply will result in increased rates of

    inflation and increased inflationary expectations. This will cause investors to increase the

    inflation premium on all debt securities, and the entire yield curve would rise; that is, all rates

    would be higher.

    Chapter 6: Interest Rates Integrated Case 119

    6-7 a. S&Ls would have a higher level of net income with a “normal” yield curve. In this situation their

    liabilities (deposits), which are short-term, would have a lower cost than the returns being

    generated by their assets (mortgages), which are long-term. Thus, they would have a positive

    “spread.”

    b. It depends on the situation. A sharp increase in inflation would increase interest rates along

    the entire yield curve. If the increase were large, short-term interest rates might be boosted

    above the long-term interest rates that prevailed prior to the inflation increase. Then, since

    the bulk of the fixed-rate mortgages were initiated when interest rates were lower, the

    deposits (liabilities) of the S&Ls would cost more than the returns being provided on the

    assets. If this situation continued for any length of time, the equity (reserves) of the S&Ls

    would be drained to the point that only a “bailout” would prevent bankruptcy. This has indeed

    happened in the United States. Thus, in this situation the S&L industry would be better off

    selling their mortgages to federal agencies and collecting servicing fees rather than holding

    the mortgages they originated.

    6-8 Treasury bonds, along with all other bonds, are available to investors as an alternative investment to common stocks. An increase in the return on Treasury bonds would increase the appeal of these bonds relative to common stocks, and some investors would sell their stocks to buy T-bonds.

     This would cause stock prices, in general, to fall. Another way to view this is that a relatively riskless investment (T-bonds) has increased its return by 4 percentage points. The return demanded on riskier investments (stocks) would also increase, thus driving down stock prices. The exact relationship will be discussed in Chapter 8 (with respect to risk) and Chapters 7 and 9 (with respect to price).

    6-9 A trade deficit occurs when the U.S. buys more than it sells. In other words, a trade deficit occurs when the U.S. imports more than it exports. When trade deficits occur, they must be financed, and the main source of financing is debt. Therefore, the larger the U.S. trade deficit, the more the U.S. must borrow, and as the U.S. increases its borrowing, this drives up interest rates.

    120 Integrated Case Chapter 6: Interest Rates

    Solutions to End-of-Chapter Problems

6-1 a. Term Rate Interest Rate 10 6 months 5.1% (%)

     1 year 5.5 8 2 years 5.6

     3 years 5.7 6

     4 years 5.8 4 5 years 6.0

    10 years 6.1 220 years 6.5 030 years 6.3 051015202530 Years to Maturity

    b. The yield curve shown is an upward sloping yield curve.

    c. This yield curve tells us generally that either inflation is expected to increase or there is an

    increasing maturity risk premium.

    d. It would make sense to borrow long term because each year the loan is renewed interest

    rates are higher. This exposes you to rollover risk. If you borrow for 30 years outright you

    have locked in a 6.3% interest rate each year.

6-2 T-bill rate = r* + IP

     5.5% = r* + 3.25%

     r* = 2.25%.

6-3 r* = 3%; I = 2%; I = 4%; I = 4%; MRP = 0; r = ?; r = ? 123T2T3

    r = r* + IP + DRP + LP + MRP.

    Since these are Treasury securities, DRP = LP = 0.

    r = r* + IP. T22

    IP = (2% + 4%)/2 = 3%. 2

    r = 3% + 3% = 6%. T2

    r = r* + IP. T33

    IP = (2% + 4% + 4%)/3 = 3.33%. 3

    r = 3% + 3.33% = 6.33%. T3

6-4 r = 6%; r = 8%; LP = 0.5%; DRP = ? T10C10

    r = r* + IP + DRP + LP + MRP.

    r = 6% = r* + IP + MRP; DRP = LP = 0. T101010

    Chapter 6: Interest Rates Integrated Case 121

     = 8% = r* + IP + DRP + 0.5% + MRP. rC101010

    Because both bonds are 10-year bonds the inflation premium and maturity risk premium on both

    bonds are equal. The only difference between them is the liquidity and default risk premiums.

    r = 8% = r* + IP + MRP + 0.5% + DRP. But we know from above that r* + IP + MRP = C101010

    6%; therefore,

     r = 8% = 6% + 0.5% + DRP C10

    1.5% = DRP.

6-5 r* = 3%; IP = 3%; r = 6.2%; MRP = ? 2T22

     r = r* + IP + MRP = 6.2% T222

     r = 3% + 3% + MRP = 6.2% T22

    MRP = 0.2%. 2

6-6 r* = 5%; I = 16%; MRP = DRP = LP = 0; r = ? 1-44

    r = r. 4RF

    r = (1 + r*)(1 + I) 1 RF

     = (1.05)(1.16) 1

     = 0.218 = 21.8%.

6-7 r = 5%; r = 6%; r = ? T11T1T2 2(1 + r) = (1.05)(1.06) T22(1 + r) = 1.113 T2

     1 + r = 1.055 T2

     r = 5.5%. T2

6-8 Let X equal the yield on 2-year securities 4 years from now:

    426 (1.07)(1 + X) = (1.075) 2(1.3108)(1 + X) = 1.5433 1/21.5433 1 + X = 1.3108(

     X = 8.5%.

    6-9 r = r* + IP + MRP + DRP + LP.

    r* = 0.03.

    IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035.

    MRP = 0.0005(6) = 0.003.

    DRP = 0.

    LP = 0.

    r = 0.03 + 0.035 + 0.003 = 0.068 = 6.8%. T7

    122 Integrated Case Chapter 6: Interest Rates

    6-10 Basic relevant equations:

     = r* + IP + DRP + MRP + IP. rttttt

But here IP is the only premium, so r = r* + IP. ttt

IP = Avg. inflation = (I + I + . . .)/N. t12

We know that I = IP = 3% and r* = 2%. Therefore, 11

r = 2% + 3% = 5%. r = r + 2% = 5% + 2% = 7%. But, T1T3T1

r = r* + IP = 2% + IP = 7%, so T333

IP = 7% 2% = 5%. 3

We also know that I = Constant after t = 1. t

We can set up this table:

     r* I Avg. I = IP r = r* + IP tt

    1 2% 3% 3%/1 = 3% 5%

    2 2% I (3% + I)/2 = IP 2

    3 2% I (3% + I + I)/3 = IP r = 7%, so IP = 7% 2% = 5%. 333

IP = (3% + 2I)/3 = 5% 3

     2I = 12%

     I = 6%.

    6-11 We’re given all the components to determine the yield on the bonds except the default risk

    premium (DRP) and MRP. Calculate the MRP as 0.1%(5 1) = 0.4%. Now, we can solve for the

    DRP as follows:

    7.75% = 2.3% + 2.5% + 0.4% + 1.0% + DRP, or DRP = 1.55%.

    6-12 First, calculate the inflation premiums for the next three and five years, respectively. They are IP 3= (2.5% + 3.2% + 3.6%)/3 = 3.1% and IP = (2.5% + 3.2% + 3.6% + 3.6% + 3.6%)/5 = 3.3%. 5

     The real risk-free rate is given as 2.75%. Since the default and liquidity premiums are zero on

    Treasury bonds, we can now solve for the maturity risk premium. Thus, 6.25% = 2.75% + 3.1%

    + MRP, or MRP = 0.4%. Similarly, 6.8% = 2.75% + 3.3% + MRP, or MRP = 0.75%. Thus, 3355

    MRP MRP = 0.75% 0.40% = 0.35%. 53

    6-13 r = r* + IP + MRP + DRP + LP C88888

    8.3% = 2.5% + (2.8% 4 + 3.75% 4)/8 + 0.0% + DRP + 0.75% 8

    8.3% = 2.5% + 3.275% + 0.0% + DRP + 0.75% 8

    8.3% = 6.525% + DRP 8

    DRP = 1.775%. 8

    Chapter 6: Interest Rates Integrated Case 123

    2 = (1.03)(1 + X) 6-14 a. (1.045)

    1.092/1.03 = 1 + X

     X = 6%.

    b. For riskless bonds under the expectations theory, the interest rate for a bond of any maturity

    is

    r = r* + average inflation over N years. If r* = 1%, we can solve for IP: NN

    Year 1: r = 1% + I = 3%; 11

     I = expected inflation = 3% 1% = 2%. 1

    Year 2: r = 1% + I = 6%; 12

     I = expected inflation = 6% 1% = 5%. 2

    Note also that the average inflation rate is (2% + 5%)/2 = 3.5%, which, when added to r* =

    1%, produces the yield on a 2-year bond, 4.5%. Therefore, all of our results are consistent.

    6-15 r* = 2%; MRP = 0%; r = 5%; r = 7%; X = ? 12

    X represents the one-year rate on a bond one year from now (Year 2).

     2 (1.07) = (1.05)(1 + X)

    1.1449 = 1 + X 1.05

     X = 9%.

9% = r* + I 2

    9% = 2% + I 2

    7% = I. 2

    The average interest rate during the 2-year period differs from the 1-year interest rate expected

    for Year 2 because of the inflation rate reflected in the two interest rates. The inflation rate

    reflected in the interest rate on any security is the average rate of inflation expected over the

    security’s life.

    6-16 r = r = 20.84%; MRP = DRP = LP = 0; r* = 6%; I = ? RF6

20.84% = (1.06)(1 + I) 1

     1.2084 = (1.06)(1 + I)

     1.14 = 1 + I

     0.14 = I.

    6-17 r = 5.2%; r = 6.4%; r = 8.4%; IP = 2.5%; MRP = 0. For Treasury securities, DRP = LP T5T10C1010

    = 0.

    DRP + LP = DRP + LP. r = ? 551010C5

     r = r* + IP T1010

    6.4% = r* + 2.5%

     r* = 3.9%.

    124 Integrated Case Chapter 6: Interest Rates

     r = r* + IP T55

    5.2% = 3.9% + IP 5

    1.3% = IP. 5

     r = r* + IP + DRP + LP C10101010

    8.4% = 3.9% + 2.5% + DRP + LP 1010

     2% = DRP + LP. 1010

    r = 3.9% + 1.3% + DRP + LP, but DRP + LP = DRP + LP = 2%. So, C555551010

    r = 3.9% + 1.3% + 2% C5

     = 7.2%.

    6-18 a. Years to Real Risk-Free

    Maturity Rate (r*) IP** MRP r = r* + IP + MRP T

     1 2% 7.00% 0.2% 9.20%

     2 2 6.00 0.4 8.40

     3 2 5.00 0.6 7.60

     4 2 4.50 0.8 7.30

     5 2 4.20 1.0 7.20

     10 2 3.60 1.0 6.60

     20 2 3.30 1.0 6.30 **The computation of the inflation premium is as follows:

     Expected Average Year Inflation Expected Inflation 1 7% 7.00% 2 5 6.00 3 3 5.00 4 3 4.50 5 3 4.20 10 3 3.60 20 3 3.30

    For example, the calculation for 3 years is as follows:

    7%;5%;3% = 5.00%. 3

    Thus, the yield curve would be as follows:

    Interest Rate (%)

    11.0

    10.5

    10.0

    9.5

    9.0

    8.5

    Exelon 8.0

    7.5

    7.0 ExxonMobil

    6.5 T-bonds

    Years to Maturity 0 2 4 6 8 10 12 14 16 18 20

    Chapter 6: Interest Rates Integrated Case 125

    b. The interest rate on the ExxonMobil bonds has the same components as the Treasury securities,

    except that the ExxonMobil bonds have default risk, so a default risk premium must be included.

    Therefore,

     = r* + IP + MRP + DRP. rExxonMobil

    For a strong company such as ExxonMobil, the default risk premium is virtually zero for short-

    term bonds. However, as time to maturity increases, the probability of default, although still

    small, is sufficient to warrant a default premium. Thus, the yield risk curve for the ExxonMobil

    bonds will rise above the yield curve for the Treasury securities. In the graph, the default risk

    premium was assumed to be 1.0 percentage point on the 20-year ExxonMobil bonds. The

    return should equal 6.3% + 1% = 7.3%.

    c. Exelon bonds would have significantly more default risk than either Treasury securities or

    ExxonMobil bonds, and the risk of default would increase over time due to possible financial

    deterioration. In this example, the default risk premium was assumed to be 1.0 percentage

    point on the 1-year Exelon bonds and 2.0 percentage points on the 20-year bonds. The 20-

    year return should equal 6.3% + 2% = 8.3%.

    6-19 a. The average rate of inflation for the 5-year period is calculated as:

    Average = (0.13 + 0.09 + 0.07 + 0.06 + 0.06)/5 = 8.20%. inflation rate

b. r = r* + IP = 2% + 8.2% = 10.20%. Avg.

c. Here is the general situation:

     1-Year Expected Arithmetic Average Maturity Risk Estimated

    Year Inflation Expected Inflation r* Premium Interest Rates

     1 13% 13.0% 2% 0.1% 15.1%

     2 9 11.0 2 0.2 13.2

     3 7 9.7 2 0.3 12.0

     5 6 8.2 2 0.5 10.7 . . . . . . . . . . . . . . . . . .

    10 6 7.1 2 1.0 10.1

    20 6 6.6 2 2.0 10.6

    126 Integrated Case Chapter 6: Interest Rates

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