This exercise is based on the following table, which lists interest rates on long-term investments (based on 10-year government bonds) in several countries in 2008.
How long, to the nearest year, will it take an investment in Germany(4.3%) to triple its value if the interest is compounded every three months?
I got 35 years but its wrong, don't know why.
thanks
Pt = Po*(1+r)^n.
r=(4.3%/4) / 100% = 0.01075= Quarterly % rate expressed as a decimal.
n = The # of compounding periods.
Pt = Po*(1.01075)^n = 3Po.
(1.01075)^n = 3Po/Po = 3.0.
n*Log(1.01075) = log3.
n = Log3 / Log(1.01075) = 102.74 Comp.
periods.
102.74Comp. / 4Comp/yr=25.7 yrs.=26 yrs
To find the number of years it will take for an investment to triple in value, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount or value of the investment
P = the initial principal or investment amount
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we want to find the time it takes for the investment in Germany to triple its value, with an interest rate of 4.3% compounded every three months.
Let's break down the problem step by step:
1. First, we need to find the factor by which the investment needs to grow in order to triple its value. Tripling the value means multiplying it by 3, so the factor is 3.
2. The interest rate is 4.3% per year, which is equivalent to 0.043 as a decimal.
3. The interest is compounded every three months, so the number of times interest is compounded per year is 12/3 = 4.
4. Since we know the principal amount (P) and the factor by which it needs to grow, we can rewrite the formula as:
3 = 1 + (0.043/4)^(4t)
(0.043/4)^(4t) = 3 - 1
(0.043/4)^(4t) = 2
Now, we can solve for t by taking the logarithm of both sides:
log((0.043/4)^(4t)) = log(2)
4t * log(0.043/4) = log(2)
t = log(2) / (4 * log(0.043/4))
Evaluating this expression, we find that t ≈ 158.63 years.
Therefore, it will take approximately 158 years (to the nearest year) for the investment in Germany to triple its value when the interest is compounded every three months, not 35 years as you initially calculated.