Question 1
BEEM119: Problem Set 2—Indicative Answers
1. If “junk bonds” are junk, then why would investors buy them? Junk bonds carry significant risk. The nature of risk is that the bad outcome does not have to materialize. If an investor can tolerate risk or even likes to bet, (s)he can buy risky bonds at low prices that would yield high rates of interest in case the good outcome materializes.
2. Why do U.S. Treasury bills have lower interest rates than large-denomination nego- tiable bank CDs? U.S. Treasury bills are considered (default) risk free–it’s the “safest” asset. In fact, the difference between the CDs’ yield and the treasury bill yields is called risk premium and captures that (and how much) the CDs are more risky than the treasury bills.
3. Risk premiums on corporate bonds are usually anticyclical; that is, they decrease dur- ing business cycles expansions and increase during recessions. Why? In recessions, corporations are more likely to get in financial troubles which spurs their default risk and thus raises risk premiums. (If, e.g., demand slumps, some companies can’t pay their bills on time or even go bankrupt. Other compa- nies relying on those bills to be paid, cannot collect the cash they need to pay off debts themselves; and so on.) In expansions, corporations are more likely to be capable to honour their debts so that risk premiums decrease.
4. If the yield curve suddenly becomes steeper, how would you revise your predictions of interest rates in the future? Why? A steeper yield curve means that long term interest rates are increasing. By the expectations theory, the long term interest rates are averages of the short term interest rates. I would expect the short term interest rates to rise in the future.
Question 2
Among the following, what makes you (un)happy? Why? (Assume that the quality of what you buy does not change. Only prices do due to the demand and supply condition of the market.)
1. The price of the painting he is planning to buy rises. Unhappy. 2. The price of the painting he bought rises. Happy.
3. The price of the bond he is planning to buy rises. Unhappy.
4. The price of the bond he bought rises. Happy.
5. The yield of the bond he bought rises. Unhappy. 6. The yield of the bond he bought drops. Happy.
Questions 3
Consider a TIPS with face value 100, coupon rate 5%, and maturity 2 years. Assume that inflation is 5% per year. What is the new face value, coupon rate, coupon payment, nominal yield, and real yield? The new face value is 110.25, the new coupon rate equals the original coupon rate of 5%, the coupon payment changes from 5 to 5.25 to 5.5125; the nominal yield changes from 5% to 10.25%; the real yield is unchanged at 5%.
π1 = 0.05, π2 = 0.05; fv0 = 100, fv1 = 105, fv2 = 110.25; cf1 = 5.25, cf2 = 5.5125;
The real yield is supposed to be unchanged. The nominal yield i changes as follows
fv=cf1+cf2 +fv2 1+i (1+i)2 (1+i)2 100 = 5.25 + 5.5125 + 110.25 = 5.25 + 115.7625
1+i (1+i)2 1+i (1+i)2 (1 + i)2 − 0.0525(1 + i) − 1.157625 = 0.
The positive root of this quadratic equation is 1.1025 implying that i = 0.1025 or 10.25%. The real yield is given by
2 (1 + i)2 1.10252 1.1025 1.1025 2 (1+r) = (1+π1)(1+π2) = 1.052 = 1.05 · 1.05 = 1.05
so that r = 0.05 or 5% as expected. (Recall that the point of TIPS is to insure the investor against inflation.)
Question 4
Suppose that a firm has issued a 10-year bond with face value $50,000 every decade since the 1990s, using its lands and plants as collateral. Suppose that the yield of this bond was 3% in the 1990s, 10% in the 2000s, and 3% in the 2010s. Which one of the following scenarios is the least likely?
1. Demand for bonds was high in the 1990s, it dropped in the 2000s, and it rose again in the 2010s.
2. James’ firm was rated as BBB in 1990s, was rated as AA in 2000s, and was rated as BBB in 2010s.
3. Real estate prices were rising or expected to rise in the 1990s and in the 2010s, but falling or expected to fall in the 2000s.
Question 5
Assume that the expectations theory is the correct theory of the term structure, calculate the (approximate) interest rates in the term structure for maturities from 1 to 5 years, and plot the resulting yield curves for the following paths of one-year interest rates over the next 5 years.
1. 5%, 6%, 7%, 6%, 5%. The yields for maturities of 1 to 5 years are approximately given by 5%, 5.5%, 6%, 6%, 5.8%.
2. 5%, 4%, 3%, 4%, 5%. The yields for maturities of 1 to 5 years are approximately given by 5%, 4.5%, 4%, 4%, 4.2%.
3. How would your yield curve change if people preferred shorter-term bonds over longer- term bonds? Holders of longer term bonds would require a liquidity premium that increases the slope of the yield curve.