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 ?
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To solve the math problems given, let's go step by step.
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?
Since the Actual/360 convention considers a year to have 360 days, the interest rate ℓ for a three-month horizon with 91 days can be calculated using the formula:
ℓ = (ending value - starting value) / (starting value x number of days)
ℓ = (1.03 - 1) / (1 x 91/360)
ℓ = 0.03 / 0.0025
ℓ = 12
Therefore, the interest rate ℓ is 12%.
2. If the interest rate r is expressed in the continuous-compounding convention and we treat three months as 1/4 years, what is r?
In continuous-compounding, the formula to calculate the interest rate is:
r = ln(ending value / starting value) / time in years
r = ln(1.03 / 1) / (1/4)
r = ln(1.03) / 0.25
Using a calculator, r ≈ 0.1197
Therefore, the continuously-compounded rate r is approximately 11.97%.
3. 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 growth factor can be calculated using the formula:
growth factor = (1 + ℓ/360)^(number of days)
growth factor = (1 + 4/360)^(31)
Using a calculator, the growth factor ≈ 1.010222
Therefore, the invested amount grows to approximately $1.010222.
4. 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?
The growth factor in continuous-compounding can be calculated using the formula:
growth factor = e^(r x time in years)
growth factor = e^(0.04 x 1/12)
Using a calculator, the growth factor ≈ 1.003333
Therefore, the invested amount grows to approximately $1.003333.
5. 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?
Using the Actual/360 convention, the rate ℓ can be calculated using the formula:
ℓ = (growth factor - 1) / (1 x number of days)
ℓ = (1.003333 - 1) / (1 x 31/360)
ℓ = 0.003333 / 0.086111
ℓ ≈ 0.038718
Therefore, the rate ℓ is approximately 3.87%.
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