Question

Time to Double or Triple an Investment The formula $$ t=\frac{\ln m}{n \ln \left(1+\frac{r}{n}\right)} $$ can be used to find the number of years \(t\) required to multiply an investment \(m\) times when \(r\) is the per annum interest rate compounded \(n\) times a year. (a) How many years will it take to double the value of an IRA that compounds annually at the rate of \(6 \% ?\) (b) How many years will it take to triple the value of a savings account that compounds quarterly at an annual rate of \(5 \% ?\) (c) Give a derivation of this formula.

Short answer

To double: approx. 11.90 years. To triple: approx. 22.09 years. Derivate using compound interest formula and natural logarithm.

Step-by-step solution

Understand the Formula

The given formula is \[ t = \frac{\ln m}{n \ln \left(1+\frac{r}{n}\right)} \]where: t is the time in years, m is the multiple of the original investment, r is the annual interest rate, n is the number of times the interest is compounded per year.

Solve for Doubling (Part a)

For doubling the value, set m = 2, r = 0.06, and n = 1 (annually compounding). Plug these values into the formula: \( t = \frac{\ln 2}{1 \cdot \ln \left(1+\frac{0.06}{1}\right)} \)

Calculate the Time for Doubling

Solve the equation: \( t = \frac{\ln 2}{\ln 1.06} \approx \frac{0.693}{0.0583} \approx 11.90 \) years

Solve for Tripling (Part b)

For tripling the value, set m = 3, r = 0.05, and n = 4 (quarterly compounding). Plug these values into the formula: \( t = \frac{\ln 3}{4 \cdot \ln \left(1+\frac{0.05}{4}\right)} \)

Calculate the Time for Tripling

Solve the equation: \( t = \frac{\ln 3}{4 \cdot \ln 1.0125} \approx \frac{1.098}{0.0497} \approx 22.09 \) years

Derive the Formula (Part c)

The formula can be derived from the compound interest formula. Start with the compound interest equation: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]For the investment to multiply m times, we have: \[ mP = P \left(1 + \frac{r}{n}\right)^{nt} \]Cancel out P: \[ m = \left(1 + \frac{r}{n}\right)^{nt} \]Take the natural logarithm of both sides: \[ \ln m = nt \ln \left(1 + \frac{r}{n}\right) \]Solve for t: \[ t = \frac{\ln m}{n \ln \left(1 + \frac{r}{n}\right)} \]

Key concepts

investment doubling time

Doubling your investment means that the value of the investment becomes twice its original amount. The time required to double your investment depends largely on the interest rate and how frequently it is compounded. We use the formula: \( t = \frac{\text{ln} \, 2}{\text{n ln} \, (1 + \frac{\text{r}}{n})} \) Here, the variable \(m\) is set to 2 because we want to find the time it takes to double the investment. Plug in the values for the annual interest rate \(r\) and the compounding frequency \(n\). For example, to double an IRA amount compounding annually at a rate of 6% (where \(r = 0.06\) and \(n = 1\)), we use: \( t = \frac{\text{ln} \, 2}{\text{ln} \, 1.06} \) Simplifying it further, we get: \( t \approx \frac{0.693}{0.0583} \approx 11.90 \) years
So, it will take approximately 11.9 years to double your investment in this scenario.

Understanding this concept is crucial for making precise investment predictions and formulating financial plans.

investment tripling time

Tripling your investment means the value of the investment becomes three times its original amount. We use the same formula as for doubling, but this time, we set \(m\) to 3: \( t = \frac{\text{ln} \, 3}{\text{n ln} \, (1 + \frac{\text{r}}{n})} \) For example, to triple the value of a savings account compounded quarterly at an annual rate of 5% (where \(r = 0.05\) and \(n = 4\)), we use: \( t = \frac{\text{ln} \, 3}{4 \, \text{ln} \, (1 + \frac{0.05}{4})} \) Simplifying this further, we get: \( t \approx \frac{1.098}{4 \, \text{ln} \, 1.0125} \approx \frac{1.098}{0.0497} \approx 22.09 \) years

So, it will take approximately 22.09 years for the value of this savings account to triple. This calculation is valuable for long-term financial goals and helps in understanding how compounding frequency affects the growth of investments.

compound interest formula derivation

To understand the formula better, let's derive it from the basic compound interest equation. Let’s start with the compound interest formula: \( A = P \, (1 + \frac{r}{n})^{nt} \) Here:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial sum of money).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the number of years the money is invested or borrowed for.

If we want the investment to multiply \(m\) times, then \(A = mP\): \( mP = P \, (1 + \frac{r}{n})^{nt} \) We can cancel out \(P\) from both sides: \( m = (1 + \frac{r}{n})^{nt} \) To solve for \(t\), we take the natural logarithm of both sides: \( \text{ln} \, m = nt \, \text{ln} \, (1 + \frac{r}{n}) \) Finally, rearranging the equation to solve for \(t\), we get: \( t = \frac{\text{ln} \, m}{n \, \text{ln} \, (1 + \frac{r}{n})} \)
This equation is fundamental for understanding the time-related aspect of compound interest, crucial for financial planning and investment strategies.