Deck 2: Interest Rates, Theories, and Duration As an Overall Risk Measure

Question 1

You have a 5-year investment holding horizon, would like to earn a 4% annual compound return each year, and you have a choice between two bonds.
Find (a) the Price, (b) Duration, and (c) Modified Duration for each bond, and answer the questions below:
$\bullet$ Bond 1 has a 4% annual coupon rate, $1000 maturity value, n = 5 years, YTM = 4% (pays a $40 annual coupon at the end of each year for each of the 5 years and $1,000 maturity payment at the end of year 5).
$\bullet$ Bond 2 is a zero coupon bond with a $1000 maturity value, and n = 5 years; YTM= 4%. (pays no coupons; only a $1,000 maturity payment at the end of year 5)

Answer 1

a. Price Bond 1 ___$1,000__ Price Bond 2 __ $821.93____
b. Duration Bond 1 ___4.63_______ Duration Bond 2 ___5______
c. Modified Duration Bond 1 __4.45 __ Modified Duration Bond 2 ___4.81_____
Price Bond 1 = $40 (PVIFA 4%, 5) + $1000 (PVIF 4%, 5)
= $40(4.4518) + $1000(.82193) = $178.07 + $821.93 = $1,000
(sells at Par Value since CR = YTM rate).
$$\begin{array}{l}\text { Duration Calculation: }\\\begin{array} { l c c c c } \text { Year } & { \text { CF } } & \text { PVIF } & \text { PVCF } & \text { WtedPVCF } \\1 & \$ 40 & .96154 & \$ 38.46 & \$ 38.46 \\2 & \$ 40 & .92456 & \$ 36.98 & \$ 73.96 \\3 & \$ 40 & .88900 & \$ 35.56 & \$ 106.68 \\4 & \$ 40 & .85480 & \$ 34.19 & \$ 136.77 \\5 & \$ 1,040 & .82193 & \$ 854.81 & \$ 4274.03 \\& \text { Totals } & & \$ 1,000 & \$ 4629.91\end{array}\end{array}$$ Duration = $4629.91/$1,000 = 4.63
Modified Duration = 4.63/1.04 = 4.45
Price Bond 2 = $1000/(1.04)^5 = $1000(.82193) = $821.93
Duration for Zero Coupon Bond 2 = Its Maturity = 5 years
Modified Duration Bond 2 = 5/1.04 = 4.81

Question 2

Which of the two bonds listed in problem #1 should you choose for your 5-year investment horizon to duration match to ensure your 5% annual compound return? Explain why. (Assume the same default risk for each bond.)

Answer 2

Choose Bond 2 with the duration of 5 years equal to your 5-year investment horizon. By duration matching, your investment will be immunized against interest rate risk to achieve the desired 4% annual compound yield if the zero coupon bond is held to maturity.

Question 3

a. If interest rates go up by 1%, what will be the % Change in the market value for each Bond's Price for the two Bonds in Problem #1?
b. Which of the 2 bonds has more price risk, and has more reinvestment risk? Explain why.

Answer 3

% Change in Price for Bond 1 __-4.45%___
% Change in Price for Bond 2 _-4.48%___
Calculation:
% Change in - Modified Duration x Change in Rate
Bond 1 % Change = -4.45(.01) = 0.0445 or- 4.45%
Bond 2 % Change = -4.81(.01) = -.0448 or -4.48%
B. Bond 1 has coupon payments, so has more reinvestment risk.
Bond 2 has more price risk, with a higher Modified Duration than Bond 1, so has higher risk of a capital loss if it has to be sold prior to maturity.

Question 4

a. For the Zero Coupon Bond 2 in Problem #1, what will be your annual compound yield for your 5 year holding period?
b. For the Coupon Bond 1 in Problem #, 1 what will be your annual compound yield if you hold the bond for 5 years (i.e. receive 5 years of coupon payments that are reinvested at a 3% annual rate), and received your maturity payment at the end of year 5? FVIFA = {(1 + r)^n - 1 / r}
c. Explain why you received your desired annual compound return for the 5-year holding period for Bond 2, but didn't receive your desired Annual Compound Return for Bond 1 for your 5 year holding period.

Question 5

Distinguish between a nominal versus a real interest rate.

Question 6

If a bond gives you a 2% nominal annual interest rate and the inflation rate over the year is 2%, what is the real ex post rate of return you receive?

Question 7

If an investor wants a real rate of return of 2% and expects inflation to be 2% next year, what nominal rate will the investor demand?

Question 8

Explain the loanable funds theory in your own words including a discussion of the supply and demand curves for loanable funds, the net suppliers of loanable funds and net lenders of loanable funds, and factors that affect respectively the demand and supply for loanable funds.

Question 9

If the Federal Reserve decided to reduce the money supply by engaging in open market bond purchases from the non-bank public, explain what will happen to the equilibrium interest rate in the U.S. In your discussion use the loanable funds theory to explain the effect on the equilibrium interest rate in an economy.

Question 10

Using the loanable funds theory, explain what will happen to the real interest rate in an economy if a new technology is developed whereby businesses will be demanding more loanable funds?

Question 11

Using the loanable funds theory and the demand and supply of loanable funds, explain what will happen to the real interest rate in an economy if a recession occurs?

Question 12

Suppose in the Wall Street Journal you see the following current (spot) interest rates for Treasury bonds with an upward sloping yield curve:
5-year bond rate = 1.45%; 10-year bond rate = 2.13%
a. Under the expectations theory, what is the expected 5-year bond rate (forward rate) 5 years from now? Based on your answer, what are rates expected to do (rise/fall/stay the same)? Explain why.
b. For the problem in 12a., if there is a liquidity premium of 0.10% for a 5-year bond and 0.20% for a 10-year bond, under the liquidity premium theory (adjusting for liquidity premiums incorporated in bond rates) what is the new expected 5-year bond rate 5 years from now? Based on your answer, what are rates expected to do (rise/fall/stay the same)? Explain why

Question 13

Explain how interest rates are determined under the market segmentation theory. Give examples of financial institutions that have preferences for particular types of securities.

Question 14

What is the preferred habitat theory, and how does it differ from the market segmentation theory?

Question 15

The Go Broncos Bank has the following questions it would like to ask you about its bank. The bank's balance sheet is as follows:

a. What is the Bank's Duration of Assets and Duration of Liabilities and its Duration gap (D-Gap)? What does the duration gap tell you about the bank's interest rate risk?
b. What is the expected % change in the value of equity with a rise in rates of 1%? Use an average loan rate of 5.5%.

Question 16

If you purchase a 5 year bond with a face value of $1,000, an annual coupon rate of 8%, and a YTM of 10% (the market rate for similar bonds), which of the following statements is true?

A) The bond will be selling at a discount at a price less than $1,000 (its face value), since the coupon rate < YTM.
B) The bond will be selling at its par (face) value of $1,000.
C) The bond will be selling at a premium at a price greater than $1,000 (its face value), since the coupon rate < YTM.
D) None of the above.

Question 17

If you invest $1000, and your investment is worth $1,100 next year, what was your annual compound yield on your one-year investment?

A) 8%
B) 5%
C) 10%
D) 110%

Question 18

If Glenda puts money in a bank account that offers a 1% annual rate, and inflation rate over the year is 3%, what is Glenda's real (inflation-adjusted) return?

A) About 0%
B) About 3%
C) About -2%
D) About 4%

Question 19

Reinvestment risk for a coupon bond is the risk of having to invest coupon income at a lower rate if interest rates fall, and Price risk is the risk of a capital loss if a bond has to be sold and interest rates rise. Both of these risks are measured by duration, which measures bond price sensitivity to changes in market rates that captures both types of risk.

Question 20

A zero coupon bond's duration will be equal to its maturity, since all of its cash payments come at maturity.

Question 21

What is the duration of a 4-year bond with an annual coupon rate of 10%,
Maturity value of $1,000, and yield to maturity of 8%?

A) 2.75
B) 3.51
C) 3.00
D) 4.00

Question 22

A bond with coupon payments will always have a duration less than its
maturity.

Question 23

A bond has a duration of 2.74, and the YTM for similar bonds is 8%, what is the bond's modified duration, and approximate percentage change in price for the bond if rates rise by 1%?

A) Modified duration of 3, and approximate change of -3%
B) Modified duration of 2.74, and approximate change of -2.74%
C) Modified duration of 2.49, and approximate change of -2.49%
D) Modified duration of 2.54 and approximate change of -2.54%

Question 24

Jim Goodwin bought a Zero Coupon Bond that has 6 years to
Maturity that pays no coupons, and has a $1,000 maturity value. Similar bonds are paying an 8% market rate. What is the price of the bond, and what is its duration?

A) Price of $630.20, and duration of 6 years.
B) Price of $1,000, and duration of 6 years.
C) Price of $630.20, and duration of 5 years.
D) Price of $850, and duration of 5 years.

Question 25

Ken invests in a 5-year bond that sells at its maturity value of $1,000 with its annual coupon rate of 10% equal to the YTM for similar bonds of 10%. Immediately after the purchase, rates rise and stay at 12%, and Ken sells the bond at the end of year 4.
What will be the total of the future value of the coupon payments at the end of year 4 and the price the bond will sell for bond sell for at the end of year 4, and what will be Ken's annual compound return?

A) FV of Coupons = $1,000; Bond Price Yr 4: $1000; ACY = 19%
B) FV of Coupons = $500; Bond Price Yr 4: $1,100; ACY = 49%
C) FV of Coupons = $400; Bond Price Yr 4: $920; ACY = 9.16%
D) FV of Coupons = $477.90; Bond Price Yr 4: $982.14; ACY = 9.92%

Question 26

Jack buys a 4-year zero coupon bond with a $1,000 maturity value for a Price of $683. What is Jack's annual compound yield at the end of 4 years?

A) 12%
B) 10%
C) 7%
D) 8%

Question 27

Jill buys a 5-year bond with a 10% annual coupon rate and $1,000 maturity value, and the YTM is 10%. What is the bond's duration and modified duration?

A) 4.17 and 3.79
B) 5.46 and 4.55
C) 6.00 and 5.46
D) 3.58 and 3.25

Question 28

Suppose Jill buys the 5-year bond with a 10% annual coupon rate and $1,000 maturity value, and YTM of 10%, and sells it at the end of 4 years
to meet her investment horizon of 4 years, and right after her purchase rates fall to 8%.
What will be the future value of the coupons and the price she sells bond for at the end of year 4, and what will her annual compound yield be?

A) FV of Coupons $800; Price $1000; ACY: 15.83%
B) FV of Coupons $400; Price $1000; ACY: 8.78%
C) FV of Coupons $450.60; Price $1,018.52; ACY: 10.09%
D) FV of Coupons $400; Price $1,018.52; ACY: 9.13%

Question 29

If George has an investment horizon of 5 years, which of the 2 bonds below would be best for him to immunize his investment from interest rate risk (holding other factors constant)?
Bond 1: Zero Coupon Bond with 5 years to maturity
Bond 2: Coupon Bond with 5 years to maturity

A) Neither bond
B) Coupon bond
C) Zero coupon bond

Question 30

Limitations of Duration include which of the following:

A) Assumes a flat yield curve with the same discount rate used for each cash flow each year.
B) Assumes a parallel shift in the yield curve with both short-term and long-term rates rising by the same amount when rates change.
C) For large portfolios, adjustments need to be made for convexity (given with a change in rates, capital gains are larger than capital losses which duration as a linear relationship doesn't adjust for).
D) All of the above.

Question 31

Suppose an investor wants a 4% real rate of return and expects inflation to be 4% next year, including compounding, what nominal rate should the investor demand?

A) 8.16%
B) 8%
C) 4%
D) 5%

Question 32

Suppose an investor purchases a bond with a 6.16% nominal rate, and inflation over the year is 2%, what is the real return that the investor receives?

A) 4%
B) 4.08%
C) 8.16%
D) 5%

Question 33

Under the loanable funds theory of interest rates, the equilibrium interest rate in an economy is determined by the intersection of the supply curve of loanable funds from net savers (investors) and changes in the money supply and the demand curve for loanable funds by net borrowers (corporations, governments).

Question 34

Under the loanable funds theory of interest rates, if a new technology is developed that increases the demand for funds by businesses, which of the following statements is correct?

A) The supply curve for loanable funds will shift back, intersecting with the demand curve for loanable funds at a higher equilibrium real interest rate.
B) The demand curve for loanable funds will shift back, intersecting with the supply curve for loanable funds at a lower equilibrium real interest rate.
C) The demand curve for loanable funds will shift out, intersecting with the supply curve for loanable funds at a higher equilibrium real interest rate.
D) No change in interest rates will occur.

Question 35

Under the loanable funds theory of interest rates if a central bank engages in an expansionary open markets operation to expand the money supply, which of the following statements is correct?

A) The supply curve for loanable funds will shift back, intersecting with the demand curve for loanable funds at a higher equilibrium real interest rate.
B) The supply curve for loanable funds will shift out, intersecting with the demand curve for loanable funds at a lower equilibrium real interest rate.
C) The demand curve for loanable funds will shift inward, intersecting with the supply curve for loanable funds at a lower equilibrium interest rate.
D) No change in interest rates will occur.

Question 36

Under the loanable funds theory of interest rate, which of the following statements is correct?

A) The demand curve for loanable funds is upward sloping with higher demand for funds from businesses and governments at higher rates.
B) The supply curve for loanable funds is downward sloping with a higher supply of funds supplied by investors at lower rates.
C) The demand curve for loanable funds is downward sloping with a higher demand for funds from businesses and governments at lower rates
D) The supply curve for loanable funds is upward sloping with a higher supply of funds provided by investors at higher rates
E) Both c and d

Question 37

Under the loanable funds theory, which of the following increase the demand for loanable funds by businesses and governments?

A) Government budget deficits
B) Business cycle expansion and expectations for profitable investment opportunities
C) Inflation expectations
D) All of the above

Question 38

Which of the factors below affects the supply of loanable funds?

A) Wealth in an economy
B) Risk and liquidity of bonds relative to other investments
C) Expected returns
D) Expected inflation
E) All of the above

Question 39

Bond risk premiums tend to go up during recessions, since default risk rises, and to go down during expansions when default risk for corporations fall.

Question 40

Yield curves are graphs of the relationship between the yield and time to maturity (holding other factors constant) for a homogeneous group of securities. Yield curves are typically downward sloping, but can also be flat or upward sloping.

Question 41

Under the pure expectations theory, the shape of the yield curve is determined by current short-term rates and expectations for future short-term rates (forward rates) with the interest rate on a long-term bond equal to the average of current short-term and expected short-term rates over the life of a long-term bond.

Question 42

Under the expectations theory if the current 10-year bond rate is 5.5%, and the current 5-year bond rate is 3.5%, what is the expected (forward) rate for a 5-year bond in 5 years from now?

A) 7.5%
B) 5.5%
C) 9.5%
D) 8.5%

Question 43

Under the Liquidity Premium Theory, long-term rates are the average of short-term rates including liquidity premiums where investors demand a liquidity premium for greater liquidity risk for longer-term securities. The Liquidity Premium Theory helps to explain why the yield curve is typically upward sloping.

Question 44

Under the Expectations theory if you have an upward sloping yield curve, rates are expected to fall.

Question 45

Suppose a current 6-year bond has a 6% rate and a 3-year bond has a 3% rate, and the liquidity premium (LP) for the 6-year bond is 0.20% and the LP for the 3-year bond is 0.10%, what is the expected (forward) rate for a 3-year bond issued 3 years from now under the Liquidity Premium Theory?

A) 8.77%
B) 9.08%
C) 8.87%
D) 10.11%

Question 46

The market segmentation theory suggests that there are separate markets for short-term bonds, immediate term bonds and long-term bonds, because different investors have preferences for specific markets, so rates are determined by separate supply and demand markets for the interest rate in each respective market, with little substitution across markets.

Question 47

If a bank has a duration of assets equal to 5, and a duration of liabilities of 2, and liabilities to total assets of 90%, what is the duration gap for the bank?

A) 3.2
B) 3
C) 5
D) None of the above.

Question 48

Ways banks can reduce a positive duration gap include taking on more long-term liabilities or taking on shorter term assets or variable rate assets. However, these changes will result respectively in a higher interest rate on liabilities and a lower rate on assets, since interest-rate risk is being passed on to customers.

Question 49

Duration gaps as measures of interest rate risk consider changes in volume and mix with rate changes.