The winner of a lottery receives m dollars at the end of...

The winner of a lottery receives $m$ dollars at the end of each year for $N$ years. The present value (PV) of this prize in today's dollars is $\mathrm{PV}=\sum_{i=1}^{N} m(1+r)^{-i},$ where $r$ is the interest rate. Calculate PV if $m=\$ 50,000, r=0.06,$ and $N=20 .$ What is $\mathrm{PV}$ if $N=\infty ?$

Key Concepts

Present Value

Present value is the current worth of a future sum of money or stream of cash flows given a specified rate of return. It accounts for the time value of money, meaning that receiving dollars today is more valuable than receiving the same amount in the future because of their potential earning capacity.

Discounted Cash Flow

Discounted cash flow is a valuation method used to estimate the value of an investment based on its expected future cash flows. These future cash flows are discounted back to their present value using a particular interest rate to account for risk and the time value of money.

Annuity

An annuity is a series of equal payments made at regular intervals over a period of time. In the context of present value calculations, annuities are evaluated by summing the present values of each individual payment, which involves discounting each payment back to the present using the interest rate.

Finite Geometric Series

A finite geometric series is a sum of terms in which each term after the first is found by multiplying the previous term by a common ratio. In financial models like annuities, the payments can be modeled as a geometric series where each payment is discounted by a factor that represents the interest rate over time.

Infinite Geometric Series

An infinite geometric series is the sum of an infinite number of terms that have a common ratio, which must have an absolute value less than one for the series to converge. In financial contexts, when the number of payment periods approaches infinity, the present value of these payments can be calculated using the formula for the sum of an infinite geometric series.

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