Question

Find the present value of \(\$ 1,000,000\) to be received after 20 years assuming continuous compounding at \(6 \%\).

Short answer

The present value is approximately \$301,194.

Step-by-step solution

Determine the present value formula for continuous compounding

The formula to find the present value with continuous compounding is given by: \[ PV = FV \times e^{-rt} \]where \( PV \) is the present value, \( FV \) is the future value, \( r \) is the annual interest rate, and \( t \) is the time in years.

Identify the known variables

Given:- Future Value (\( FV \)) = \$1,000,000- Annual interest rate (\( r \)) = 0.06 (6\%)- Time (\( t \)) = 20 years

Plug the variables into the formula

Substituting the given values into the formula:\[ PV = 1,000,000 \times e^{-0.06 \times 20} \]

Calculate the exponent

First, calculate the exponent part:\[ -0.06 \times 20 = -1.2 \]So, the formula now is:\[ PV = 1,000,000 \times e^{-1.2} \]

Compute the value of \( e^{-1.2} \)

Using a calculator to find \( e^{-1.2} \):\[ e^{-1.2} \approx 0.301194 \]

Calculate the present value

Now, multiply the future value by \( e^{-1.2} \):\[ PV = 1,000,000 \times 0.301194 \approx 301,194 \]So, the present value is approximately \$301,194.

Key concepts

continuous compounding

In finance, continuous compounding refers to the process of earning interest on an investment continuously, with the interest being calculated and added to the principal at every moment. This leads to the interest itself earning interest over time. The formula to calculate the present value (PV) with continuous compounding is given by: \[ PV = FV \times e^{-rt} \]. The formulas use the natural exponential function, which is represented by the constant \( e \approx 2.71828 \). Here are the key components:

• FV: Future Value; the amount you expect to receive or have in the future
• r: Annual interest rate expressed as a decimal
• t: Time in years

By using continuous compounding, you capture the effect of compounding more frequently than daily, monthly, or yearly interest, resulting in a slightly higher accumulated value over time.

future value

Future Value (FV) represents the amount of money an investment will grow to over a period of time at a specified interest rate. When dealing with continuous compounding, the future value is linked to the present value via the equation: \[ FV = PV \times e^{rt} \]. For our example, we aim to find the present value of receiving \$1,000,000 in 20 years with an annual interest rate of 6%.
Given:

• Future Value (\(FV\)) = \$1,000,000
• Annual interest rate (\(r\)) = 0.06
• Time (\t) = 20 years

Substituting these into the formula for continuous compounding yields: \[ PV = 1,000,000 \times e^{-0.06 \times 20} \]. By solving this step-by-step, we follow: 1. Calculate the exponent: \(-0.06 \times 20 = -1.2\). 2. Compute \(e^{-1.2}\), which approximately equals 0.301194. 3. Finally, multiply the future value by this exponent result: \[PV = 1,000,000 \times 0.301194 \approx 301,194 \]. Thus, in this scenario, the present value is approximately \$301,194.

exponential decay

Exponential decay describes the process of reducing an amount by a consistent percentage rate per unit of time. In the context of present value calculations with continuous compounding, exponential decay is evident in the formula \[ PV = FV \times e^{-rt} \]. The exponent \( e^{-rt} \) decays the future value (\(FV\)) back to its present value (\(PV\)) by accounting for the interest rate over time in a continuous manner.

Breaking it down:

• Exponential factor: The factor \( e^{-rt} \) decays as time (\(t\)) and the rate (\(r\)) increases.
• Decay process: For our example, we calculate \(-r \times t\) as \(-0.06 \times 20 = -1.2\). This shows the amount of decay applied to the future value over 20 years.
• Resultant value: Converting this decay factor \( e^{-1.2} \approx 0.301194 \), we see how much the future value is reduced to give us the present worth.

Exponential decay ensures that the present value reflects a realistic measure of the future sum discounted back through continuous compounding.
Understanding these concepts deepens your grasp of time value of money and the importance of interest calculations in finance.