Current year (end of year) 0

Real risk-free rate 2.5%

Expected inflation Year

1 8.0%

2 6.0%

3 4.0%

4 3.0%

5 5.0%

6 and beyond 3.0%

Maturity risk premium Annual 0.1%

Maximum 2.0%

Years Real risk-free Inflation Maturity Risk Treasury

1 2.50% 8.00% .10% 10.60%

2 2.50% 7.00% 0.20% 9.70%

3 2.50% 6.00% 0.30% 8.80%

4 2.50% 5.25% 0.40% 8.15%

5 2.50% 5.20% 0.50% 8.20%

10 2.50% 4.10% 1.00% 7.60%

20 2.50% 3.55% 2.00% 8.05%

30 2.50% 3.37% 2.00% 7.87%

To this point, we have constructed yield curves based upon hypothetical data. The first yield curve operates under the simple assumption that inflation is expected to rise in the future. To some extent, actual yield curves are constructed in similar ways. The true Treasury yield curve is determined by graphing Treasury security yields of varying maturities. Every security's yield is based upon investor attitudes and expectations regarding future market Conditions. In other words, the sort of "building" we have done with our yield curves sort of occurs implicitly for every security.

CORPORATE BONDS

Default Risk Premiums and the Liquidity Premium

The construction of corporate yields is a process of beginning with the appropriate Treasury yield curve and adding in these additional yield premiums. However, the determination of these premiums can be tricky. It seems logical that default risk on corporate bonds should somehow be a function of the firm's corporate bond rating. Apart from that we have little guidance regarding the determination of the default risk premium. However, we could assume that the relationship between corporate bonds of a given rating and Treasury securities of the same maturity would have a fairly stable relationship. By that token, we might be able to derive a good point estimator of the default risk premium by looking at the recent average default spread for different rated corporate bonds. This information is as follows:

Bond Rating Default spread

AAA 1.00%

B 2.80%

This tells us the average default spreads of corporate securities with various bond ratings. We will use this data as the starting point for our corporate yield curves. Naturally, the first question that arises is, "Does the default risk premium change, or does it always stay the same?". Logically, the idea of a time-varying default risk premium seems fairly plausible. The longer the maturity of the security, the greater the possibility of default.Therefore, we need some sort of mechanism for simulating this relationship. Just as we did for the maturity risk premium, we will "manufacture" a relationship by which the default risk premium interacts with the time to maturity.The following formula is simply made up, but it gives us a default relationship with which we are comfortable.

DRP t =Default spread * (1.02)(t-1)

With all of that having been said, we can step forward and try to construct corporate yield curves. Naturally, yield curves can be created for corporate bonds of any rating. However, we have chosen to create curves for only AAA and B rated bonds.This exercise is for purely illustrative purposes, so rather than complicate the graph with a lot of curves, we will create two curves to show the relationship between yield curves.

Real Maturity, AAA AAA B-rated, B-rated

Years, risk-free, Inflation, Risk Treasury, DRP, bond, DRP, Bond

1 2.50% 8.00% 0.10% 10.60% 1.00% 11.60% 2.80% 13.40%

2 2.50% 7.00% 0.20% 9.70% 1.02% 10.72% 2.86% 12.56%

3 2.50% 6.00% 0.30% 8.80% 1.04% 9.84% 2.91% 11.71%

4 2.50% 5.25% 0.40% 8.15% 1.06% 9.21% 2.97% 11.12%

5 2.50% 5.20% 0.50% 8.20% 1.08% 9.28% 3.03% 11.23%

10 2.50% 4.10% 1.00% 7.60% 1.20% 8.80% 3.35% 10.95%

20 2.50% 3.55% 2.00% 8.05% 1.46% 9.51% 4.08% 12.13%

30 2.50% 3.37% 2.00% 7.87% 1.78% 9.64% 4.97% 12.84%

Looking at the yield curve we have constructed, we see a relationship that we should have expected. We see that at any length to maturity, the yield on corporate bonds is always greater than the yield on Treasuries. This is logical because corporate securities carry a default risk, and Treasuries do not. Furthermore, we observe that at any length to maturity the corporate security with the lower rating always has a higher yield than a corporate bond with a higher rating. Once again, this is a logical conclusion. Remember, a bond rating, among other things, gives you an indication of the expected possibility of default. Naturally, this possibility is greater for bonds with lower ratings, and this possibility of default gives the indicated security greater risk. Greater risk should result in a higher yield.

1. In Year 6 and thereafter, inflation is expected to be 3 percent. The maturity risk premium (MRP) is 0.1 percent per year to maturity for bonds with maturities greater than six months, with a maximum MRP equal to 2 percent. The real risk-free rate of return is currently 2.5 percent, and it is expected to remain at this level long into the future. Compute the interest rates on Treasury securities with maturities equal to one year, two years, three years, four years, five years, 10 years, 20 years, and 30 years. Draw the yield curve.
2. Discuss the yield curve that is constructed from part A.

1. Rework part (a) assuming one year has passedthat is, today is January 1 of Year 2. All the other information given in part (a) is the same. Rework part (a) again assuming two, three, four, and five years have passed.

1. Assume that all the information given previously is the same and the default risk premium for corporate bonds rated AAA is 1.5 percent whereas it is 4 percent for corporate bonds rated B. Compute the interest rates on AAA-and B-rated corporate bonds with maturities equal to one year, two years, three years, four years, five years, 10 years, 20 years, and 30 years.

Answer rating: 100% (QA)

To compute the interest rates on Treasury securities and corporate bonds with different maturities we need to consider the given information and formu... View the full answer