What is the APY for money invested at each rate? (A) 6% compounded monthly (B) 4% compoumded continuously
APY = (1 + r)^n - 1.
A. r = APR / 365 = rate per compounding
period expressed as a decimal.
n = 365 days=the number of compounding
periods.
APY = (1 + 0.00016438)^365 - 1,
= 1.061830 - 1 = 0.061830 = 6.183%
Correction:
P = Po(1+r)^n
A. Let Po = $1.00 @ 6% for 1 year.
r = (6%/12)/100% = 0.005 = Monthly% rate
expressed as a decimal.
n = 1comp/mo. * 12mo. = 12 Compounding
periods.
P = 1(1.005)^12 = 1.06168.
I = P - Po = 1.06168 - 1.000 = $0.06168
APY = (I/Po)*100% = (0.06168/1.0)*100% =
6.168%.
B. P = Po*e^(rt)
r = (4%/100% = 0.04 = APR expressed as a decimal.
t = 1 yr.
rt = 0.04/yr. * 1yr = 0.04
P = 1.0*e^0.04 = $1.04081
I = 1.04081 - 1.00 = $0.04081
APY = (I/Po)*100% = 0.04081/1.00)*100% =
4.081%
To find the Annual Percentage Yield (APY) for money invested at a given rate, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
(A) For 6% compounded monthly:
First, we need to convert the annual interest rate to a decimal form, so 6% becomes 0.06.
Let's assume we invest $1000:
P = $1000
r = 0.06 (6% in decimal form)
n = 12 (compounded monthly)
t = 1 (1 year)
A = $1000(1 + 0.06/12)^(12*1)
Simplifying the equation:
A = $1000(1.005)^12
Calculating:
A ≈ $1061.68
The final amount after one year is approximately $1061.68.
(B) For 4% compounded continuously:
Again, let's assume we invest $1000:
P = $1000
r = 0.04 (4% in decimal form)
n = infinity (compounded continuously)
t = 1 (1 year)
The formula for continuously compounded interest is:
A = P * e^(rt)
A = $1000 * e^(0.04*1)
Calculating:
A ≈ $1040.81
The final amount after one year is approximately $1040.81.
Therefore, the APY for money invested at each rate is approximately:
(A) 6% compounded monthly: $1061.68
(B) 4% compounded continuously: $1040.81
To calculate the Annual Percentage Yield (APY) for money invested at different rates, we need to use the appropriate formulas for each compound interest scenario.
For scenario (A), where the interest is compounded monthly at a rate of 6%, we can use the formula:
APY = (1 + (r/n))^n - 1
where:
r is the annual interest rate (as a decimal),
n is the number of compounding periods per year.
In this case, the annual interest rate is 6%, which is equal to 0.06 in decimal form. The compounding periods per year are 12 (since it is compounded monthly).
Plugging the values into the formula, we have:
APY = (1 + (0.06/12))^12 - 1
Using a calculator, we can solve this equation and find the APY for scenario (A).
For scenario (B), where the interest is compounded continuously at a rate of 4%, we can use the formula:
APY = e^(r) - 1
where e is the mathematical constant approximately equal to 2.71828.
In this case, the annual interest rate is 4%, which is equal to 0.04 in decimal form.
Using the formula, we have:
APY = e^(0.04) - 1
Again, using a calculator, we can calculate the value of e^(0.04) and subtract 1 to find the APY for scenario (B).