Actual/360 Interest Rate Convention
In this assignment, you will solve problems on Interest Rate Conventions.
Suppose an investment of $1 made today will be worth $1.03 in three months.
1. If the interest rate ℓ is expressed in the Actual/360 convention and the three-month horizon has 91 days in it, what is ℓ ?
If the interest rate r is expressed in the continuous-compounding convention and we treat three months as 1/4 years, what is r ?
Consider an investment of $1 over a horizon of one month.
If the interest rate ℓ expressed in the Actual/360 convention is 4% and the one-month horizon has 31 days in it, to what does the invested amount grow to?
If you had to express the same outcome using a continuous-compounding convention, and we treat one month as 1/12 of a year, what is the continuously-compounded rate r ?
Consider an investment of $1 over a horizon of one month.
If the interest rate r expressed in the continuously–compounded terms is 4% and we treat the one month horizon as 1/12 of a year, to what does the invested amount grow?
If you had to express the same outcome using an Actual/360 convention and the one month horizon has 31 days in it, what is the rate ℓ ?
Problem 1: Actual/360 Interest Rate Convention
To find the interest rate ℓ in the Actual/360 convention with a three-month horizon of 91 days, we can use the formula:
ℓ = (Future Value - Present Value) / (Present Value * Number of Days / 360Given that the Present Value (PV) is $1, the Future (F) is $1.03, and Number of Days is 91, we can substitute these values the formula:
ℓ = ($1.03 - $) / ($1 * 91/360)
Simpl,
=0.03 / ($1 * 0.2528)
ℓ ≈ 0.7 or 11.87%
Therefore, the interest rate ℓ in the Actual/360 convention is approximately 11.87%.
Problem 2: Continuous-Compounding Interest Rate Convention
To find the continuous-compounding rate r with a three-month horizon treated as 1/4 years, we can use the formula:
r = ln(Future Value / Present Value) / Time
Given that the Present Value (PV) is $1, the Future Value (FV) is $1.03, and the Time is 1/4 years, we can substitute these values into the formula:
r = ln($1.03 / $1) / (1/4)
Simplifying,
r = ln(1.03) / 0.25
Using a calculator,
r ≈ 0.1197 or 11.97%
Therefore, the continuous-compounding rate r is approximately 11.97%.
Problem 3: Actual/360 Interest Rate Convention
To find the value to which the investment grows over a one-month horizon using the Actual/360 convention, we can use the formula:
Future Value = Present Value * (1 + ℓ * Number of Days / 360)
Given that the Present Value (PV) is $1, the interest rate ℓ is 4%, and the Number of Days is 31, we can substitute these values into the formula:
Future Value = $1 * (1 + 0.04 * 31/360)
Simplifying,
Future Value = $1 * (1 + 0.0114)
Future Value ≈ $1.0114
Therefore, the investment grows to approximately $1.0114.
Problem 4: Continuous-Compounding Interest Rate Convention
To find the continuously-compounded rate r to achieve the same outcome as in Problem 3, we can use the formula:
r = ln(Future Value / Present Value) / Time
Given that the Present Value (PV) is $1, the Future Value is $1.0114, and the Time is 1/12 years, we can substitute these values into the formula:
r = ln($1.0114 / $1) / (1/12)
Simplifying,
r = ln(1.0114) / (1/12)
Using a calculator,
r ≈ 0.0095 or 0.95%
Therefore, the continuously-compounded rate r is approximately 0.95%.
Problem 5: Actual/360 Interest Rate Convention
To find the interest rate ℓ in the Actual/360 convention with a one-month horizon of 31 days, we can use the formula:
ℓ = (Future Value - Present Value) / (Present Value * Number of Days / 360)
Given that the Present Value (PV) is $1, the Future Value is $1.0114 (from Problem 3), and the Number of Days is 31, we can substitute these values into the formula:
ℓ = ($1.0114 - $1) / ($1 * 31/360)
Simplifying,
ℓ = $0.0114 / ($1 * 0.0861)
ℓ ≈ 0.1316 or 13.16%
Therefore, the interest rate ℓ in the Actual/360 convention is approximately 13.16%.