The present value P V of a capital asset that provides a...

The present value $P V$ of a capital asset that provides a perpetual stream of revenue that flows continuously at a rate of $R(t)$ dollars per year is given by $$ P V=\int_{0}^{\infty} R(t) e^{-r t} d t $$ where $r$, expressed as a decimal, is the annual rate of interest compounded continuously. (a) Find the present value of an asset if it provides a constant return of $\$ 100$ per year and $r=8 \%$. (b) Find the present value of an asset if it provides a return of $R(t)=1000+80 t$ dollars per year and $r=7 \%$

The present value $P V$ of a capital asset that provides a perpetual stream of revenue that flows continuously at a rate of $R(t)$ dollars per year is given by $$ P V=\int_{0}^{\infty} R(t) e^{-r t} d t $$ where $r$, expressed as a decimal, is the annual rate of interest compounded continuously.
(a) Find the present value of an asset if it provides a constant return of $\$ 100$ per year and $r=8 \%$.
(b) Find the present value of an asset if it provides a return of $R(t)=1000+80 t$ dollars per year and $r=7 \%$

Key Concepts

Improper Integration of Discounted Cash Flows

When the cash flows occur continuously over an infinite horizon, the present value calculation is transformed into an improper integral. This involves integrating the product of the flow function R(t) and the continuously compounded discount factor e^(–rt) from zero to infinity. Solving this integral requires techniques from calculus, such as evaluating standard integral forms and using integration by parts when the cash flow function is time-dependent.

Perpetuity

A perpetuity refers to a financial asset or cash flow stream that continues indefinitely with no end date. The valuation of a perpetuity involves summing an infinite series of cash flows, each discounted back to the present. When the cash flow is constant and the discounting is done correctly, the present value of a perpetuity is simply the constant cash flow divided by the discount rate, provided that the discount rate is positive.

Continuous Compounding

Continuous compounding involves the process of applying interest at every instant, as opposed to discrete compounding periods. In the context of present value calculations, the cash flows are discounted using an exponential decay function, e^(–rt), where r is the continuously compounded rate and t is time. This approach is particularly useful when dealing with cash flows that accrue continuously over time.

Present Value and Discounting

The present value concept is used to determine the current worth of future cash flows by taking into account the time value of money. Cash flows that occur in the future are discounted back to the present using an appropriate discount rate, which reflects the opportunity cost of capital and the risks associated with the investment. This adjustment helps in making the cash flows comparable in today's monetary terms.

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