Question

In 1626 Peter Minuit, governor of the colony of New Netherland, bought the island of Manhattan from Indians paying with beads, cloth, and trinkets worth \(\$ 24\). Find the value of this sum in year 2000 at \(5 \%\) compounded a) continuously and b) annually.

Short answer

Continuously: \$3178636651.2; Annually: \$178947177.11

Step-by-step solution

Define Continuous Compound Interest Formula

The formula for continuous compound interest is: \(P(t) = P_0 \times e^{rt}\) Where \(P(t)\) is the amount of money accumulated after time \(t\), \(P_0\) is the initial principal balance (\$24), \(r\) is the rate of interest (5\% or 0.05), and \(t\) is the time in years (2000 - 1626 = 374 years).

Calculate Continuous Compound Interest

Using the formula defined, plug in the values: \(P(t) = 24 \times e^{0.05 \times 374}\) Compute the exponent first: \(0.05 \times 374 = 18.7\) Therefore, the formula becomes \(P(t) = 24 \times e^{18.7}\) Use a calculator to find \(e^{18.7}\), which approximately equals 132442618.8. Thus, the value is \(P(t) = 24 \times 132442618.8 = 3178636651.2\)

Define Annual Compound Interest Formula

The formula for annual compound interest is: \(A = P \times \big(1 + \frac{r}{n}\big)^{nt}\) Where \(A\) is the amount of money accumulated after time \(t\), \(P\) is the initial principal balance (\$24), \(r\) is the rate of interest (5\% or 0.05), \(n\) is the number of times interest is compounded per year (1 for annual), and \(t\) is the time in years (2000 - 1626 = 374 years).

Calculate Annual Compound Interest

Using the formula defined, plug in the values: \(A = 24 \times \big(1 + \frac{0.05}{1}\big)^{1 \times 374}\) Simplify the expression inside the parenthesis: \(1 + 0.05 = 1.05\) The formula then becomes \(A = 24 \times 1.05^{374}\) Use a calculator to find \(1.05^{374}\), which approximately equals 7456132.38. Thus, the value is \(A = 24 \times 7456132.38 = 178947177.11\)

Key concepts

continuous compounding

Continuous compounding means that interest is added to the principal balance an infinite number of times in a given time period. This method uses the mathematical constant e (approximately 2.71828).

When compounding is continuous, even small increases in interest rates can have significant effects over long periods.

Formula: The continuous compounding formula is:
\[P(t) = P_0 \times e^{rt}\]
In this formula,

• \(P(t)\) is the amount at time \(t\),
• \(P_0\) is the initial principal,
• \(e\) is approximately 2.718,
• \(r\) is the annual interest rate,
• \(t\) is the number of years.

Using our example, \(P_0 = 24\), \(r = 0.05\), and \(t=374\). Plugging these values in, we get:
\[P(t) = 24 \times e^{0.05 \times 374} = 24 \times e^{18.7}\]
Evaluating \(e^{18.7}\), we get approximately \(132442618.8\). Thus, \(P(t) = 24 \times 132442618.8 = 3178636651.2\).

annual compounding

Annual compounding means that interest is added to the principal balance once per year. This type of compounding provides a straightforward way to calculate interest.
Formula: The annual compounding formula is:
\[A = P \times \big(1 + \frac{r}{n}\big)^{nt}\]
In this formula,

• \(A\) is the amount at time \(t\),
• \(P\) is the initial principal,
• \(r\) is the annual interest rate,
• \(n\) is the compounding frequency (1 for annual),
• \(t\) is the number of years.

Using the same example, \(P=24\), \(r=0.05\), \(n=1\), and \(t=374\). Plugging these values in, we get:
\[ A = 24 \times \big(1 + \frac{0.05}{1}\big)^{1 \times 374} = 24 \times 1.05^{374}\]
Evaluating \(1.05^{374}\), we get approximately \(7456132.38\). Thus, \(A = 24 \times 7456132.38 = 178947177.11\).

financial mathematics

Financial mathematics covers various formulas and concepts needed to understand how money grows over time. It is crucial for making informed financial decisions.

Key concepts include:

• Compound Interest: Interest calculated on the initial principal and also on the accumulated interest of previous periods.
• Simple Interest: Interest calculated only on the initial principal.
• Present Value: The current worth of a sum of money to be received in the future, considering a specific interest rate.
• Future Value: The value of a current asset at a future date, considering a specific interest rate.

For Peter Minuit's Manhattan purchase, understanding these concepts helps us see how the value of money changes over centuries. Using financial mathematics principles, we can calculate future values using both continuous and annual compounding methods.

interest rate calculation

Interest rate calculation is fundamental in understanding how financial products work. It is crucial to know how to compute it whether it's for loans, savings, or investments.
Interest rates can be calculated in two main ways:

• Simple Interest: Calculated using the formula: \[SI = P \times r \times t\], where \(SI\) is the simple interest, \(P\) is the principal amount, \(r\) is the rate of interest, and \(t\) is the time period.
• Compound Interest: There are two methods of compounding interest - continuous and discrete (like annual compounding).

Choosing the right method depends on the context.
For instance, if we use continuous compounding to find out how much the initial $24 in 1626 would be worth in year 2000 at a 5% interest rate, we get a different result compared to annual compounding. Understanding these calculations helps in making smarter financial decisions.