Problem 5 - Applied Calculus

Question

Assuming an interest rate of \(5 \%\) compounded continuously, (a) Find the future value in 6 years of a payment of \(\$ 12,000\) made today. (b) Find the future value of an income stream of \(\$ 2000\) per year over 6 years. (c) Which is larger, the future value from the lump sum in part (a) or from the income stream in part (b)? Explain why this makes sense financially.

Step-by-Step Solution

Answer

The lump sum is larger at $16,198.32 compared to $13,395.48 for the income stream.

Step 1: Understand compounded interest formula

For continuously compounded interest, the future value (FV) is calculated using the formula: \[ FV = PV \times e^{rt} \]where \(PV\) is the present value, \(r\) is the annual interest rate, and \(t\) is the time in years.

Step 2: Calculate future value of lump sum payment

Substitute the values \(PV = 12000\), \(r = 0.05\), and \(t = 6\) into the formula for continuously compounded interest:\[ FV = 12000 \times e^{0.05 \times 6} \]Calculate to find:\[ FV \approx 12000 \times e^{0.3} \approx 12000 \times 1.34986 \approx 16198.32 \]

Step 3: Understand continuous income stream formula

For an income stream with continuous compounding, the future value is calculated as:\[ FV = \int_{0}^{t} Ce^{r(t-x)} \, dx \]where \(C\) is the constant income per year, and \(x\) is the variable of integration representing the time of each payment.

Step 4: Calculate future value of income stream

Substitute the values \(C = 2000\), \(r = 0.05\), and \(t = 6\) into the income stream formula:\[ FV = \int_{0}^{6} 2000 \times e^{0.05(6-x)} \, dx \]This simplifies to:\[ FV = 2000 \times e^{0.3} \int_{0}^{6} e^{-0.05x} \, dx \]Calculate the integral:\[ FV = 2000 \times e^{0.3} \left[ -\frac{1}{0.05} e^{-0.05x} \right]_{0}^{6} \]\[ FV = 2000 \times e^{0.3} \times (-20) \left( e^{-0.3} - e^{0} \right) \]\[ FV = 2000 \times 1.34986 \times (-20) \left( 0.74082 - 1 \right) \]\[ FV \approx 2000 \times 1.34986 \times (-20) \times -0.25918 \approx 13395.48 \]

Step 5: Compare future values and explain

The future value of the lump sum calculated in part (a) is approximately \(16198.32\), while the future value of the income stream calculated in part (b) is approximately \(13395.48\).The lump sum is larger. This makes financial sense because the entire amount is invested upfront, allowing more time for compound interest to accrue compared to smaller annual payments spread over time.

Key Concepts

Future Value

The future value (FV) refers to the worth of a current asset or sum of money at a specific date in the future, given a particular rate of return. This concept is crucial in financial planning and helps in understanding how much an investment today will grow over time.
The future value is affected by the interest rate, the duration of investment, and the method of compounding.

• A higher interest rate or a longer investment period results in a larger future value.
• Future value also plays a role in evaluating alternatives such as income streams or lump sum investments.

Continuous Compounding

Continuous compounding calculates interest where the frequency of compounding is theoretically infinite.
In this method, interest accrues constant addition, leading to the highest potential growth compared to other compounding frequencies, such as yearly or monthly.
The formula to compute future value using continuous compounding is:

• \( FV = PV \times e^{rt} \)
• Where \( e \) is Euler's number, approximately 2.71828.

For example, on a lump sum of $12,000 with a5% interest rate compounded continuously over 6 years, the future value grows significantly. Compared to periodic compounding, continuous compounding can maximize the returns on investment.

Income Stream

An income stream refers to a series of payments over time, often received in intervals such as annually or monthly.
The future value of an income stream considers what these payments will be worth at some point in the future considering a specific interest rate.

• The integration approach is commonly used to calculate the future value when the income is continuous.
• The formula typically accounts for each payment occurring at different times, with each amount compounding until maturity.

In financial contexts, income streams may originate from salary, dividends, or rental income, each impacting the overarching financial model applied to prospectively calculate their worth.

Lump Sum Payment

A lump sum payment is a single, substantial amount paid at one time, rather than broken into installments. This form of payment allows the entire principal to start growing from day one.
In scenarios involving compound interest, investing a lump sum upfront usually results in a higher future value compared to smaller, staggered contributions over time.

• This is because the whole amount earns interest cumulatively over the entire period.
• When compared to income streams, lump sum investments yield more significant growth due to the uninterrupted compounding effect.

For instance, in the problem provided, the lump sum yields a higher future value than the income stream due to its upfront investment, highlighting the power of early and comprehensive investment in yielding financial growth.