A saving bank offer 1000 pesos certificate of deposit. Each certificate can be redeemed for 2000 pesos after 8.5 years. What is the nomiral annual interest rate if the interest is compounded monthly?
1000(1 + r/1200)^(12*8.5) = 2000
r = 8.18%
To find the nominal annual interest rate compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = The final amount (2000 pesos)
P = The principal amount (1000 pesos)
r = The annual interest rate (what we need to find)
n = The number of times interest is compounded per year (12, since it is compounded monthly)
t = The number of years (8.5 years)
Plugging in the given values, we have:
2000 = 1000(1 + r/12)^(12 * 8.5)
Now let's solve for r. First, divide both sides by 1000:
2 = (1 + r/12)^(12 * 8.5)
Next, take the natural logarithm (ln) of both sides to remove the exponent:
ln(2) = ln[(1 + r/12)^(12 * 8.5)]
lns(x^y) = y * ln(x), so we have:
ln(2) = (12 * 8.5) * ln(1 + r/12)
Now, divide both sides by (12 * 8.5) and simplify:
ln(2)/(12 * 8.5) = ln(1 + r/12)
Finally, solve for r by raising e (the base of natural logarithms) to the power of both sides:
e^(ln(2)/(12 * 8.5)) = 1 + r/12
Subtract 1 from both sides and multiply by 12:
r = (e^(ln(2)/(12 * 8.5)) - 1) * 12
Evaluating this expression, we can find the nominal annual interest rate.
To calculate the nominal annual interest rate when the interest is compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (2000 pesos)
P = the principal amount (1000 pesos)
r = the annual interest rate (what we need to find)
n = the number of times interest is compounded per year (12 times for monthly)
t = the number of years (8.5 years)
Plugging in the given values, we have:
2000 = 1000(1 + r/12)^(12 * 8.5)
To find the value of r, we can rearrange the equation and solve for it. Here are the steps:
1. Divide both sides of the equation by 1000:
2000/1000 = (1 + r/12)^(12 * 8.5)
2. Simplify the left side of the equation:
2 = (1 + r/12)^(12 * 8.5)
3. Take the natural logarithm (ln) of both sides:
ln(2) = ln[(1 + r/12)^(12 * 8.5)]
4. Apply the logarithmic property to bring down the exponent:
ln(2) = (12 * 8.5) * ln(1 + r/12)
5. Divide both sides by (12 * 8.5):
ln(2)/(12 * 8.5) = ln(1 + r/12)
6. Apply the inverse natural logarithm (e^x) to both sides:
e^(ln(2)/(12 * 8.5)) = 1 + r/12
7. Subtract 1 from both sides:
e^(ln(2)/(12 * 8.5)) - 1 = r/12
8. Multiply both sides by 12:
12 * (e^(ln(2)/(12 * 8.5)) - 1) = r
Now we can use a calculator to evaluate the right side of the equation. This will give us the nominal annual interest rate, r.