(a) Find the present and future value of an income stream of...

(a) Find the present and future value of an income stream of $\$ 6000$ per year for a period of 10 years if the interest rate, compounded continuously, is $5 \%$
(b) How much of the future value is from the income stream? How much is from interest?

Key Concepts

Continuous Compounding

This concept refers to the way interest is compounded infinitely often, resulting in the exponential growth of an amount over time. The main formula involves the exponential function (e^(rt)), which provides a more precise model of accumulation or discounting compared to discrete compounding methods.

Present Value

Present value is the concept of determining what a future stream of cash flows is worth in today’s dollars. When interest is compounded continuously, the present value is calculated by discounting future cash amounts using the exponential decay factor (e^(-rt)), thus allowing for an accurate assessment of their current worth.

Future Value

Future value is the accumulation of present values and periodic payments over time, expressed as the total amount in the future after earning interest. For continuously compounded interest, the future value is obtained by applying the exponential growth factor (e^(rt)) to the present values or periodic payments, providing a measure of the total accumulated wealth at the end of the period.

Annuity

An annuity is a series of equal payments or receipts over a set period of time. In the context of continuous compounding, the valuation of an annuity involves integrating the continuous income stream discounted or compounded at an exponential rate, offering a more refined calculation than traditional discrete methods.

Separation of Contributions

This concept involves distinguishing between the principal (the actual cash flows or income stream) and the interest earnings accrued over time. In financial analysis, isolating these components helps in understanding how much of the future value is due to the original payments versus the growth generated by the interest.

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