Final answer:
The price of a three-month $13,000 T-bill with a simple annual discount rate of 4.17% would be approximately $12,861.56, and the actual interest rate paid by the Treasury when annualized would be approximately 4.36%.
Step-by-step explanation:
Calculating the Price and Interest Rate of a Treasury Bill
To calculate (a) the price of the Treasury bill (T-bill) and (b) the actual interest rate paid by the Treasury for a three-month $13,000 T-bill with a simple annual discount rate of 4.17%, we need to use the formula for the price of a Treasury bill which takes into account the discount rate and the time to maturity. The formula is:
Price = Face Value / (1 + (discount rate * (days to maturity / 365)))
To find the price for a $13,000 T-bill with a 4.17% annual discount rate for 3 months (or approximately 90 days), we plug in the values:
Price = $13,000 / (1 + (0.0417 * (90 / 365)))
Price = $13,000 / (1 + 0.010260274)
Price = $13,000 / 1.010260274
Price = $12,861.56 (approximately)
The actual interest rate paid by the Treasury is not the same as the discount rate. It is calculated based on the price paid for the T-bill and the face value, and it assumes that the interest is paid at the maturity, so it's a simple interest calculation:
Interest = Face Value - Price
Interest = $13,000 - $12,861.56
Interest = $138.44
To get the actual annualized interest rate, we convert the interest to an annual basis:
Actual interest rate = (Interest / Price) * (365 / days to maturity)
Actual interest rate = ($138.44 / $12,861.56) * (365 / 90)
Actual interest rate = 0.010763 * 4.0555556
Actual interest rate = 4.36% (approximately)