Estimating Interest Rate Using Risk Premiums
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MINI-CASE 1: Estimating Interest Rate Using Risk Premiums. [5 points]
(References: See essentially Schedule of Classes Units 1, 3 and 6)
- Preliminary Remarks:
- The Mini-Case Assignment:
The purpose of this mini-case is to demonstrate how the required rate of return on a fixed-income security can be determined, using the risk premium perspective. The mini-case brings together in a simple model several of the various risks associated with fixed-income securities. From a corporation perspective, the nominal interest rate to be determined in this mini-case can serve as a basis for the firm’s before-tax cost of debt. At the issue of the security, if the nominal interest rate is set as the coupon rate that is equal to the yield to maturity (so that the price equals the par value), then the rate can also be interpreted as the discount rate. From the perspective of investors (buyers of the security), the rate could be interpreted as the investor’s nominal required rate of return.
- Your company, Binghamton Truck, Inc., is about to offer a new issue of corporate bonds to the investing marketplace. You have been asked by your CFO to provide a reasonable estimate of the nominal interest rate (nominal yield), R_{d}, for a new issue of Aaa-rated bonds to be offered by Binghamton Truck.
- Some agreed-upon procedures related to generating estimates for key variables in the relevant equation, R_{d} = R^{*}_{rf} + IRP + DRP + MP + LP, are as follows:
- The current (mid-2008) financial market environment is considered representative of the prospective tone of the market near the time of offering the new bonds to the investing public. This means that current interest rates will be used as benchmarks for some of the variable estimates. All estimates will be rounded off to hundredths of a percent; thus, 6.288 becomes 6.29 percent.
- The real risk-free rate of interest, R^{*}_{rf}, is the difference between the calculated average yield on 3-month Treasury bills and the inflation rate.
- The inflation-risk premium, IRP, is the rate of inflation expected to occur over the life of the bond under consideration.
- The default-risk premium, DRP, is estimated by the difference between the average yield on Aaa-rated bonds and 30-year Treasury bonds.
- The maturity premium, MP, is estimated by the difference between the calculated average yield on 30-year Treasury bonds and 3-month Treasury bills.
- Binghamton Truck’ bonds will be traded on the New York Exchange for Bonds, so the liquidity premium, LP, will be slight. It will be greater than zero; however, because the secondary market for the firm’s bonds is more uncertain than that of some other truck producers, it is estimated at 3 basis points.
Note: A basis point is one one-hundredth of 1 percent. (E.g., 1 basis point = 0.01%; 25 basis points = 0.25%)
- Based on your research, the mid-2008 estimates of the representative interest and inflation rates are as follows: (1) 3-Month T-Bills = 4.89%, (2) 30-Year T-Bonds = 5.38% (use this as proxy for 20-year T-Bonds), (3) Aaa-Rated Corporate Bonds = 6.24%, and (4) Inflation Rate = 3.60%. Visit online Federal Reserve Bank of St. Louis (Google “Federal Reserve Bank of St. Louis FRED”) and update the above data with the most recently available rates for each of the above fixed income securities and for the inflation rate.
- Required Task: Complete the Solution Table below, which is presented in form of a formula required to determine R_{d}. Place your answers (values) in the cells below the variables in the second row, and show your calculations below the Table, where applicable, of how you obtained the value for each of the variables. Similarly, use your most recent collected rates to complete the third row of the worksheet below. Briefly comment on the differences between the two results (i.e. results obtained from above old data versus results obtained from recent data you collected).
Solution to Mini-Case 1 (show your work below the table, as appropriate):
R*_{rf} |
+ IRP |
+ DRP |
+ MRP |
+ LRP |
= |
R_{d} |
Source of data for part II of Mini-Case 1:
FRED – ECONOMIC DATA, by The Federal Reserve Bank of St. Louis
https://fred.stlouisfed.org/categories
Look for the following data, under either “Prices” or “Interest Rates”:
Inflation Rate: PCETRIM12M159SFRBDAL
PCETRIM12M159SFRBDAL |
Trimmed Mean PCE Inflation Rate, Percent Change from Year Ago, Monthly, Seasonally Adjusted |
Adjusted |
Three (3)-Month Treasury Bill
TB3MS |
3-Month Treasury Bill: Secondary Market Rate, Percent, Monthly, Not Seasonally Adjusted |
30-Year Treasury Constant Maturity Rate
DGS30 |
30-Year Treasury Constant Maturity Rate, Percent, Daily, Not Seasonally Adjusted |
20-Year Treasury Constant Maturity Rate
DGS20 |
20-Year Treasury Constant Maturity Rate, Percent, Daily, Not Seasonally Adjusted |
Moody’s Seasoned Aaa Corporate Bond Yield
AAA |
Moody’s Seasoned Aaa Corporate Bond Yield, Percent, Monthly, Not Seasonally Adjusted |
MINI-CASE 2: BOOTSTRAPPING METHOD FOR ESTIMATING SPOT RATES. [5 points]
- Introduction: As discussed in the text and the PowerPoints, the Yield Curve should be constructed using theoretical rates, which represent the spot rates corresponding to each maturity. These rates and corresponding calculated prices are referred to as “Spot Rates” (or pure discount rate) and “Implied Zeroes” (or zero-coupon prices). The purpose of this mini-case is to illustrate how to calculate these implied prices and spot rates. Ideally, the calculated spot rates are the rates that should be used to discount the respective cash flows, instead of using for example a single yield to maturity when determining the value of a bond.
- The Problem: Consider the problem of finding the pure discount bond prices from the coupon prices that are available. Table 1 gives data for three bonds for a period of three years.
Table 1: Coupon Bond Prices and Coupon Payments. (Notice that the Cash Flows in years 1 through Year 3 already reflect the coupon payments and the par value at maturity.)
Bond |
Price |
Year 1 |
Year 2 |
Year 3 |
1 2 3 |
99.50 101.25 100.25 |
105 6 7 |
0 106 7 |
0 0 107 |
- Let Pi be the price of bond i (e.g., if i=1, then Pi=P1 for bond 1). Let ci denote the dollar coupon associated with bond i, and y1 as a one-year spot rate of interest. Then, we can denote the price of the first bond as
- Required task: Use the above procedure, called “Bootstrapping” to complete Table 2 below for maturities 2 and 3.
Then, using known data, solve for y1.
Finally, we can solve for the one-year implied zero price, z1, as follows:
Note: z1 is also referred to as the “discount factor” or the present value of a $1 to be received at time t in the future. The two terms to the right of the z1 equation are equivalent ways of finding the answer. They should give the same answer.
Following the same procedure, we can solve iteratively for y2 and z2, etc ….
Note that, after solving for yi in one step it becomes a known value in the next step. For example, when solving for y2 and z2, P1, P2 and y1 are known values that you can just plug in the formula for P2 to find y2 and then solve for z2.
(Note: I provided the answers for maturity 1, for illustration. However, show your work, i.e., show the steps you followed to reach the answers, including your work to verify the answers already provided).
Table 2: Implied Zeroes and Spot Rates. (Answers for maturity 1 are already given to you for illustration. This will not count in your grade.)
Maturity (Years) |
Implied Zero (zi) |
Spot Rate (yi) |
1 |
z1 = 0.9476 |
y1 = 5.53% |
2 |
z2 = ? |
y2 = ? |
3 |
z3 = ? |
Y3 = ? |
1 day ago
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