Question

Manhattan Island is said to have been bought by Peter Minuit in 1626 for \(\$ 24\). Suppose that Minuit had instead put the \(\$ 24\) in the bank at \(6 \%\) interest compounded continuously. What would that \(\$ 24\) have been worth in 2000 ?

Short answer

The $24 would be worth approximately $130.9 billion in 2000.

Step-by-step solution

Understand Continuous Compounding Formula

The formula for continuously compounded interest is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (\$24 in this case), \( r \) is the annual interest rate (0.06 for 6%), and \( t \) is the time the money is invested for.

Calculate Time Period

The time period \( t \) is the difference between the year 2000 and the year 1626, which is \( 2000 - 1626 = 374 \) years.

Plug Values Into the Formula

Substitute \( P = 24 \), \( r = 0.06 \), and \( t = 374 \) into the formula \( A = Pe^{rt} \).\[A = 24 \times e^{0.06 \times 374}\]

Calculate the Exponent

Compute the exponent part \( 0.06 \times 374 = 22.44 \).

Compute \( e^{22.44} \) Using a Calculator

Calculate \( e^{22.44} \). This requires a calculator and will give approximately \( 5.454 imes 10^{9} \).

Compute Final Amount

Now substitute back to find \( A \).\[A = 24 \times 5.454 \times 10^9 = 1.309 imes 10^{11}\]Thus, the amount would be approximately \( 1.309 \times 10^{11} \) dollars.

Key concepts

Exponential Growth

Exponential growth describes a situation where a quantity increases at a rate proportional to its current value. This means the larger the quantity gets, the faster it grows. In the context of finance, this is often seen with methods of interest that compound continuously. Here, the interest is not added at fixed intervals (like monthly or annually) but is calculated constantly. This can lead to significant growth over long periods. The formula to calculate exponential growth in continuous compounding is given as:- \( A = Pe^{rt} \) where:- \( A \) = final amount- \( P \) = principal or initial amount- \( r \) = rate of intereste = Euler's number (approximately 2.718)- \( t \) = time in years- Using this formula allows us to understand how investments grow exponentially over time.

Interest Calculation

Interest calculation is essential for understanding how money grows over time in a financial scenario. When we talk about interest, we're referring to the cost of borrowing money or the reward for investing it. Interest can be calculated in different ways, with continuous compounding being one of the most powerful. In continuous compounding, interest is calculated constantly and added back to the balance without interval breaks. This leads to more significant returns in the long run because the interest itself earns interest on a continuous basis. To calculate the interest in the given scenario: - First, identify the key variables: principal amount (P = \( 24 \)), interest rate (r = 0.06), and time (t = 374 years).- Substitute these into the formula \( A = Pe^{rt} \).- Finding the value of \( e^{22.44} \) helps to demonstrate the power of continuous growth.- Continuous compounding can grow investments to massive amounts if allowed to run over long periods, as shown by the final calculation of \( 1.309 \times 10^{11} \) dollars.

Historical Value Comparison

Comparing historical values help us understand the immense impact of time on money's growth. If an amount like \( 24 \) dollars was invested in 1626 with continuous compounding at a 6% rate, it would expand to an astounding \( 1.309 \times 10^{11} \) dollars by the year 2000.This transformation demonstrates the effect of both time and exponential growth.- Small amounts of money, given enough time, can accumulate to monumental sums.- It's the combination of a consistent interest rate and the long duration that turns a modest principal into a huge end amount.- Historical comparison illustrates why time is one of the most critical factors in finance, emphasizing the significance of long-term investments.Such comparisons not only shed light on potential financial growth but also offer fascinating insights into the value of money through different historical periods, helping us understand both the absolute and relative growth of wealth.