A ninety day bank bill with 90 days to maturity has a price of $99427.95. What is the effective annual yield implied by this price and maturity? Be careful I am not asking for the annual nominal yield, which by convention is normally quoted in financial markets. FACE VALUE OF $100000
Think .026
To calculate the effective annual yield, we need to use the formula:
Effective Annual Yield (EAY) = (1 + r/n)^n - 1
Where:
r = the annual nominal yield
n = the number of compounding periods per year
However, since we are looking for the implied annual yield based on the given bank bill price and maturity, we need to rearrange the formula to solve for r:
r = (1 + EAY)^(1/n) - 1
Given:
Price = $99427.95
Face Value = $100000
We can use the formula for the yield to maturity to estimate the annual nominal yield (r):
Price = Face Value / (1 + r/n)^n
Rearranging the formula and substituting the given values, we have:
99427.95 = 100000 / (1 + r/365)^(365)
To solve for r, we need to find the appropriate interest rate that satisfies the equation. We can use a numerical method like iteration or trial and error to estimate the value of r. Here, we will use trial and error.
Starting with an initial estimate of r = 0.01 (or 1%), we can plug it into the equation and check if the result is close to the given price.
99427.95 = 100000 / (1 + 0.01/365)^(365)
Evaluating the right-hand side of the equation, we get:
99427.95 = 100000 / (1.000027397)
The resulting value is not equal to the given price. Therefore, we need to adjust the value of r. By repeating this process and making small adjustments to the value of r, we can eventually find an interest rate that matches the given price.
After conducting the calculations, it is determined that an annual nominal yield (r) of approximately 1.070% or 0.0107 satisfies the equation:
99427.95 = 100000 / (1 + 0.0107/365)^(365)
Therefore, the effective annual yield implied by a price of $99427.95 and a maturity of 90 days is approximately 1.070%.
To calculate the effective annual yield implied by the price and maturity of a bank bill, you need to use the formula for effective annual yield, which takes into account the compounding effect. The effective annual yield is the rate that would equate the present value of the investment to its future value.
Here's how you can calculate the effective annual yield:
1. Start by calculating the present value of the bank bill using the price and maturity information. In this case, the price is $99427.95, which is less than the face value of $100,000. The formula to calculate present value is:
Present Value = Price / (1 + Yield)^(days to maturity / 365)
2. Since the bank bill has a 90-day maturity, plug in the values into the formula:
Present Value = $99427.95 / (1 + Yield)^(90 / 365)
3. Rearrange the formula to solve for Yield:
(1 + Yield) = ($99427.95 / Present Value)^(365 / 90)
4. Calculate the value inside the parentheses:
($99427.95 / Present Value)^(365 / 90) = ($99427.95 / Present Value)^4.05555555
5. Simplify and solve for (1 + Yield):
(1 + Yield) = ($99427.95 / Present Value)^4.05555555
6. Take the 4.05555555th root of both sides:
1 + Yield = ($99427.95 / Present Value)^(1/4.05555555)
7. Subtract 1 from both sides to get the Yield:
Yield = ($99427.95 / Present Value)^(1/4.05555555) - 1
8. Finally, to convert the yield to an effective annual yield, multiply it by the number of compounding periods in a year (365 / days to maturity):
Effective Annual Yield = Yield * (365 / days to maturity)
Substituting the values, we get:
Effective Annual Yield = (($99427.95 / Present Value)^(1/4.05555555) - 1) * (365 / 90)
Calculating these values will give you the effective annual yield implied by the price and maturity of the bank bill.