Assuming an interest rate of 5 % compounded continuously,...

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.

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.

Key Concepts

Continuous Compounding

This concept refers to the process of calculating interest where the frequency of compounding is infinite, leading to the use of the natural exponential function. It models the growth of an investment by applying the formula using the continuous growth factor e^(rt), where r is the annual interest rate and t is the time in years.

Exponential Accumulation Factor

The exponential accumulation factor, given by e^(rt) in continuous compounding, is used to determine the future value of money by accounting for the effect of interest applied at every moment. This factor captures the essence of exponential growth, where values increase at a rate proportional to their current amount.

Future Value of a Lump Sum

Calculating the future value of a one-time payment involves multiplying the initial principal by the exponential growth factor, reflecting how money grows over time under continuous compounding. This concept highlights the time value of money, showing how present funds become more valuable in the future.

Future Value of a Continuous Income Stream

When dealing with a stream of continuous payments, the future value is computed by integrating each individual payment’s growth over the period until the future date. This approach sums the effects of continuous cash inflows, each compounded from the time of receipt to the target date, thereby effectively capturing the cumulative impact of repeated, continuously compounded investments.

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