n March 2007, the Mega Millions lottery had a $390 million jackpot, the largest in the world at the time.  The lottery winners had the choice of taking the payments over time (an Annual Payout or annuity) or taking a single lump sum payment (the Cash Option).  The annual payout was $15 million every year for 26 years.  The Cash Option was to receive an immediate payment of $233 million. 

If you were the winner, what would you do, take the Annual Payout or the Cash Option?

Answering this question requires understanding the time value of money and taxes.  In this topic we will ignore taxes and instead focus on the time value of money as captured by interest rates and annuities.

An annuity is a series of fixed payments over time, such as the Annual Payout from a lottery.

There are two types of annuities: ordinary annuities and annuities in-advance.   In an ordinary annuity, you receive the payments at the end of the period, e.g. at the end of the year.  With an annuity in-advance, you receive the payment at the beginning of the period.  Lottery payouts are annuities in-advance; if you choose the Annual Payout option, you get the first payment right away, the next payment one year later, and so on.

The present value of the Annual Payout is given by the annuity formula (a derivation is in the appendix to this chapter).  The formula says if you receive a constant payment C for t periods and the interest rate is r, then the present value P(C,t), of an ordinary annuity is