Suppose that the 6-month US Treasury bill rate is equal to 5% and the forward rate on a 6-month Treasury bill 6 months from now is 7.24%. (both in yearly terms). What is the rate of a 1-year bill?
To find the rate of a 1-year Treasury bill, we can start by understanding the relationship between short-term rates and future rates. The forward rate is an estimate of what the interest rate will be in the future. In this case, the 6-month Treasury bill 6 months from now has a forward rate of 7.24%.
Now, let's break down the problem step by step:
1. Convert all rates to semi-annual (6-month) rates:
- The 6-month Treasury bill rate is given as 5% per year. To convert it to a semi-annual rate, we divide by 2, giving us 2.5% every 6 months.
- The forward rate of the 6-month Treasury bill in 6 months is given as 7.24% per year. Dividing by 2, we get 3.62% every 6 months.
2. Calculate the one-year Treasury bill rate using the relationship between short-term rates and future rates:
- If we assume that the rates for the two 6-month periods are the same, we can calculate the 1-year rate by compounding the 6-month rate twice (once for each 6-month period).
- Using the formula for compound interest: 1 + r = (1 + r1) * (1 + r2), where r is the 1-year rate, r1 is the 6-month rate for the first period, and r2 is the 6-month rate for the second period.
- Plugging in the values: 1 + r = (1 + 0.025) * (1 + 0.0362)
- Solving the equation, we find: r ≈ 0.06294 or 6.294%
Therefore, the rate of a 1-year Treasury bill is approximately 6.294% per year.