Interest Rate Conventions
Sample Solution
The Actual/360 convention is a day-count convention that assumes there are 360 days in a year. So, a three-month period would have 91 days.
The interest rate ℓ is calculated as follows:
ℓ = (1 + r)^(360/t) -
Full Answer Section
where:
- r is the interest rate per year
- t is the number of days in the period
In this case, we have:
ℓ = (1 + r)^(360/91) - 1
We can solve for r as follows:
r = (1.03)^(91/360) - 1
≈ 0.037
So, the interest rate ℓ is 0.037, or 3.7%.
If the interest rate r is expressed in the continuous-compounding convention and we treat three months as 1/4 years, what is r ?
The continuous-compounding convention is a day-count convention that assumes interest is compounded continuously. So, the interest earned in a period is calculated as the product of the principal and the interest rate raised to the power of the number of days in the period.
The interest rate r is calculated as follows:
r = ln(1 + ℓ)
where:
- ℓ is the interest rate expressed in the Actual/360 convention
In this case, we have:
r = ln(1.037)
≈ 0.037
So, the interest rate r is also 0.037, or 3.7%.
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?
Using the Actual/360 convention, the interest earned on an investment of $1 for one month at a rate of 4% is calculated as follows:
Interest = ℓ * Principal * Days/360
Interest = 0.04 * 1 * 31/360
≈ $0.01
So, the invested amount grows to $1 + $0.01 = $1.01.
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 ?
Using the continuous-compounding convention, the interest earned on an investment of $1 for one month at a rate of 4% is calculated as follows:
Interest = r * Principal * Time
Interest = 0.04 * 1 * 1/12
≈ $0.0033
So, the invested amount grows to $1 + $0.0033 = $1.0033.
The continuously-compounded rate r is calculated as follows:
r = ln(1 + Interest/Principal)
r = ln(1.0033)
≈ 0.033
So, the continuously-compounded rate r is also 0.033, or 3.3%.
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?
Using the continuously-compounded rate r, the invested amount of $1 for one month at a rate of 4% grows to:
$1 * e^{r * Time}
$1 * e^{0.04 * 1/12}
≈ $1.0033
So, the invested amount grows to $1.0033.