I have been staring at this problem forever, and cant seem to dig it up in my book. Please help!
Two competing bank are trying to attract customers.
(a) Ally Bank has an account which earns 25% interest every 10 years. Assuming the interest is compounded weekly, find both the nominal and effective annual interest rates.
(b) Patriot Bank has a continuously compounded account with 3% interest. If 100 dollars is deposited in the account at Ally Bank and 70 is deposited in the account at Patriot Bank, exactly how long will it take for the money in the two accounts to be equal?
N = (1 + E)^12 -1
where N is the nominal interest rate per year; and E is the effective monthly interest rate per year.
For Ally Bank,
(N+1)^10 = 1.25
Use this equation to solve for N:
N+1 = log10(1.25)
N = log10(1.25) - 1
Then use the first equation to find E
To solve these problems, we need to use some financial formulas and concepts. Let's break it down step by step.
(a) Ally Bank:
The nominal annual interest rate refers to the stated rate of interest per year. In this case, the nominal rate is 25% every 10 years.
To find the nominal interest rate compounded weekly, we can use the formula:
Nominal Rate = ((1 + r/n)^(n*t)) - 1
where:
- r is the annual interest rate (in decimal form)
- n is the number of times the interest is compounded per year
- t is the number of years
In this case, r = 0.25 (since the nominal rate is 25%),
n = 52 (since the interest is compounded weekly),
t = 1 (since we want the annual rate).
Plugging in these values:
Nominal Rate = ((1 + 0.25/52)^(52*1)) - 1
Nominal Rate ≈ 0.2637 or 26.37%
The effective annual interest rate takes compounding into account. We can use the formula:
Effective Annual Rate = (1 + r/n)^n - 1
Using the values r = 0.25 (as before) and n = 52:
Effective Annual Rate = (1 + 0.25/52)^52 - 1
Effective Annual Rate ≈ 0.2677 or 26.77%
So, the nominal annual interest rate for Ally Bank is 26.37%, and the effective annual interest rate is 26.77%.
(b) Patriot Bank:
The continuous compounding formula is used to calculate the value in a continuously compounded account over time:
Value = P * e^(r*t)
where:
- P is the principal amount (initial deposit)
- r is the continuously compounded interest rate (in decimal form)
- t is the time in years
- e is Euler's number (~2.71828)
In this case, we have $100 deposited in the Ally Bank account (P = 100) and $70 in the Patriot Bank account.
To find the time it takes for the two accounts to be equal, we equate the two formulas:
100 * e^(0.25/10 * t) = 70 * e^(0.03 * t)
Simplifying the equation:
e^(0.025t) = 0.7 * e^(0.03t)
Taking the natural logarithm (ln) of both sides:
0.025t = ln(0.7) + 0.03t
Rearranging the equation:
0.005t = ln(0.7)
Solving for t:
t = ln(0.7) / 0.005
Using a calculator:
t ≈ 165.64 years
So, it will take approximately 165.64 years for the money in the two accounts to be equal.
Remember, these calculations involve financial formulas and would typically require a financial calculator or software.