Question

A person places Rs. 20000 in a saving account which pays 5 percent interest per annum, compounded continuously. Find (a) the amount in the account after three years, and \((\mathrm{b})\) the time required for the account to double in value, presuming no withdrawals and no additional deposits. \((\ln 2=0.693)\)

Short answer

Answer: After three years, the amount in the account is approximately Rs. 23,236. It takes approximately 13.86 years for the account to double in value, with no withdrawals and no additional deposits.

Step-by-step solution

Part a: Calculating the amount in the account after three years

To calculate the amount in the account after three years, we will use the continuous compounding formula \(A = P * e^{(rt)}\). The given values are P = 20000, r = 5% (0.05), and t = 3 years. We substitute the values into the formula and solve for A: \(A = 20000 * e^{(0.05 * 3)}\) Now, calculate the amount A: \(A ≈ 20000 * e^{0.15} ≈ 20000 * 1.1618\) \(A ≈ 23,236\) After three years, the amount in the account is approximately Rs. 23,236.

Part b: Calculating the time required for the account to double in value

To find the time required for the account to double in value, we will again use the continuous compounding formula \(A = P * e^{(rt)}\). We want to double the principal amount, which means that \(A = 2 * P\), and we are given that the interest rate is r = 5% (0.05). We need to find the time t. We can set up the formula like this: \(2 * P = P * e^{(0.05 * t)}\) Divide both sides by P to eliminate it: \(2 = e^{(0.05 * t)}\) To solve for t, we can take the natural logarithm (ln) of both sides: \(\ln(2) = \ln(e^{(0.05 * t)})\) Using the property of logarithms that \(\ln(e^x) = x\), we have: \(0.693 = 0.05 * t\) Now, solve for t: \(t = \frac{0.693}{0.05} ≈ 13.86\) It takes approximately 13.86 years for the account to double in value, with no withdrawals and no additional deposits.

Key concepts

Continuous Compounding Interest Formula

Understanding how your investments grow over time is crucial, and the continuous compounding formula is integral to that understanding. Continuous compounding interest represents the scenario where interest is calculated and added to the principal balance without a break—in essence, an infinite number of times per period.

Mathematically, the formula is expressed as: \[A = P * e^{(rt)}\] where:\begin{itemize}\item \(A\) is the final amount in the account.\item \(P\) stands for the principal amount (the initial sum or loan).\item \(r\) represents the annual interest rate (as a decimal).\item \(t\) indicates the time in years.\item \(e\) is the base of the natural logarithm, approximately equal to 2.71828.\end{itemize}
Applying this formula, as seen in the exercise, allows for a precise calculation of future investment value under the assumption of continuously compounded interest. It provides a more accurate result than compounding annually, semi-annually, or quarterly because it takes interest accumulation at each possible instant into account.

Natural Logarithm (ln)

The natural logarithm, abbreviated as 'ln', is a fundamental concept in mathematics, particularly when dealing with exponential functions and growth processes. The natural logarithm is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828.

The key property that makes \(ln\) so useful in compounding is that \(ln(e^x) = x\), which allows us to 'undo' the effect of an exponential function. This property is particularly handy when solving for time \(t\) or the rate \(r\) in continuous compounding scenarios, as demonstrated in the exercise provided.

As an aside, 'ln' allows for the transformation of multiplicative relationships into additive ones, thus simplifying many complex growth-related calculations. The visual simplicity of \(ln(2) = 0.693\) hides its power in enabling us to solve the time it takes for an investment to double under continuous compounding, seen in part (b) of the exercise.

Time Value of Money

The 'time value of money' is a key concept in finance that suggests money available now is worth more than the same amount in the future due to its potential earning capacity. This core principle underpins the idea of interest and the preference for receiving money sooner rather than later.

Essentially, it encompasses the opportunity costs associated with waiting for money. It is a concept employed to determine what an investment made today will be worth in the future, adjusting for factors such as inflation and interest earned.

Applying the time value of money concept with continuous compounding, like in our exercise, highlights its practical relevance. It allows us to calculate not just the future value of a present sum, but also the time necessary for an investment to reach a particular goal under constant growth, offering a comprehensive view of both the 'when' and 'how much' aspects of investment growth.