Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account.
A $20 comma 000 deposit in an account with an APR of 2.5%.
I will do the middle one, you do the rest
amount = 20 e^(5*.025)
= 20 e^.125 = 22.66
What does APY stand for ?
To compute the balance in the account after a certain period of time using continuous compounding, we can utilize the formula:
A = P * e^(rt)
Where:
A = the balance in the account after time t
P = the principal amount (initial deposit)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form)
t = the time period in years
Given:
P = $20,000
r = 2.5% = 0.025 (converting APR to decimal)
Now, let's calculate the balance in the account after 1, 5, and 20 years.
For 1 year:
t = 1 year
A = $20,000 * e^(0.025 * 1)
To calculate this, we need the value of e^(0.025), which is approximately 1.0253.
So, A = $20,000 * 1.0253 = $20,506.00 (rounded to the nearest cent)
For 5 years:
t = 5 years
A = $20,000 * e^(0.025 * 5)
Again, we need the value of e^(0.125), which is approximately 1.1331.
So, A = $20,000 * 1.1331 = $22,662.00 (rounded to the nearest cent)
For 20 years:
t = 20 years
A = $20,000 * e^(0.025 * 20)
Now, we need the value of e^(0.5), which is approximately 1.6487.
So, A = $20,000 * 1.6487 = $32,974.00 (rounded to the nearest cent)
Next, let's find the Annual Percentage Yield (APY) for the account. The APY represents the actual rate earned or paid over a year, taking into account compounding.
The formula to calculate APY using the APR (Annual Percentage Rate) is:
APY = (1 + (r/n))^n - 1
Where:
r = the annual interest rate (in decimal form)
n = the number of compounding periods per year
Given:
APR = 2.5% = 0.025 (converting to decimal)
n = continuous compounding (as mentioned in the question)
Using the formula, we can calculate the APY:
APY = (1 + (0.025/1))^1 - 1 = 0.025 or 2.5%
Therefore, the APY for this account would be 2.5%.