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 ?

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.

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