First bank pays 7 percent simple interest on its investment accounts. If second bank pays interest on its accounts compounded annual, what rate should the bank set if it wants to match first bank over an investment horizon of 10 years?
Let the investment be $100 (or any amount you feel like)
At simple interest:
Amount = 100 + 100*.07*10 = 170
Compound interest: let the rate be i
100(1+i)^10 = 170
(1+i)^10 = 1.7
take 10th root
1+i = 1.7(1/10) = appr 1.0545
so the rate is appr 5.45%
Nepali transalate
Alternative solution
Different method
Ah, the eternal contest of banks! Well, if the first bank pays a simple interest of 7 percent, the second bank needs to step up its game to match it over a 10-year investment horizon. Since compound interest is the name of the game, let's calculate the equivalent rate. Now, please don't blame me if these numbers make your head spin faster than a merry-go-round!
To match the first bank over 10 years, the second bank needs to solve this compound interest puzzle. We can use the formula:
A = P(1 + r/n)^(nt),
where A is the final amount, P is the principal (initial amount), r is the interest rate, n is the number of compounding periods per year, and t is the number of years.
Since we want to match the 7 percent, we can plug some values into this equation. Let's pretend the principal P is $1000 (just for fun). Now, the second bank would need to determine the interest rate (r) it should set to match the first bank.
After rearranging a bit and plugging in the values:
(1 + r/n)^(nt) = A/P,
(1 + r/n)^(10n) = 1.07.
Now, now, before despairing over the math, let me interject with a silly joke to lighten the mood:
Why don't scientists trust atoms? Because they make up everything!
Alright, back to business! Since we can't solve this equation directly, we can approximate it using trial and error or leave it to a fancy financial calculator. After some number crunching, it turns out that the second bank would need to set its interest rate (compounded annually) at approximately 6.63% to match the 7% simple interest rate of the first bank over a 10-year investment horizon.
Remember, this is just an approximation, and actual calculations may vary. But hey, at least we had a bit of fun along the way, right? Now go forth and conquer the world of banking with a smile!
To find the equivalent interest rate for the second bank that matches the 7 percent simple interest rate of the first bank over a 10-year investment horizon, we need to calculate the compound interest rate.
First, let's define the variables:
- Principal amount (P) = $1 (Assuming an initial investment of $1 for simplicity)
- Simple interest rate (r1) = 7% per year
- Time period (t) = 10 years
- Compound interest rate (r2) = ?
The formula for calculating the future value (FV) using compound interest is:
FV = P(1 + r2)^t
Since the principal amount is $1, we can rewrite the formula as:
FV = (1 + r2)^t
We want the future value (FV) after 10 years to match the simple interest earned by the first bank, which is 7% of the principal amount. So the future value should be:
FV = (1 + 0.07)P = (1 + 0.07)
Plugging this value into the formula, we have:
(1 + r2)^t = 1 + 0.07
To isolate r2, we take the t-th root of both sides:
1 + r2 = (1 + 0.07)^(1/t)
Now, subtract 1 from both sides to solve for r2:
r2 = (1 + 0.07)^(1/t) - 1
Substituting the values, we get:
r2 = (1.07)^(1/10) - 1
Calculating this expression, we find that the compound interest rate (r2) should be approximately 0.0677 or 6.77% for the second bank to match the first bank's 7% simple interest rate over a 10-year investment horizon.