6. (TCO F) Suppose the December CBOT Treasury bond futures contract has a quoted price of 80-07. What is the implied annual interest rate inherent in the futures contract? Assume this contract is based on a 20-year Treasury bond with semiannual interest payments. The face value of the bond is $1,000, and the semiannual coupon payments are $30. The annual coupon rate on the bonds is $60 per bond (or 6%). The futures contract has 100 bonds.
To calculate the implied annual interest rate inherent in the futures contract, we need to follow a series of steps:
1. Determine the current market price of the bond: The quoted price of 80-07 represents a price of 80 and 7/32 of a point. To convert this to decimal form, we divide 7 by 32 and add it to the whole number. In this case, the decimal price is 80.2188.
2. Calculate the cash settlement value: The cash settlement value is calculated by multiplying the bond's face value ($1,000) by the quoted price in decimal form (80.2188). This gives us a cash settlement value of $80,218.80.
3. Determine the invoice price: The invoice price is calculated by deducting the accrued interest from the cash settlement value. The accrued interest is calculated as the fraction of the semiannual coupon payments ($30) for the remaining period until maturity. Since there are 20 years to maturity, or 40 six-month periods, the accrued interest is (40 - 0) * ($30 / 2) = $600.
4. Calculate the implied annual interest rate: The implied annual interest rate can be determined by using the following formula: Implied annual interest rate = (100 - invoice price) / (invoice price * conversion factor). The conversion factor is calculated as: conversion factor = (bond's coupon rate * time to maturity) / (1 + bond's yield).
Using the given information, the coupon rate is 6%, the time to maturity is 20 years, and the bond's yield needs to be determined. We can solve for the bond's yield by assuming a reasonable initial value (e.g., 5%) and using trial and error or an iterative method to converge to the correct yield.
Let's assume that the bond's yield is 5%. Plugging these values into the conversion factor formula, we get:
Conversion factor = (0.06 * 20) / (1 + 0.05) = 0.63.
Now, we can calculate the implied annual interest rate:
Implied annual interest rate = (100 - invoice price) / (invoice price * conversion factor)
= (100 - $80,218.80) / ($80,218.80 * 0.63)
= 0.2509, or 25.09%.
Therefore, the implied annual interest rate inherent in the futures contract is 25.09%.