Lottery Lump Sum vs Annuity
With the US Powerball and Mega Millions lotteries, you don't actually just get the "amount you won". You can either get your winnings paid out in 30 installments over the next 29 years or you can settle for a bit over 52% of your winnings. There's no universal right answer for which is better, but with some math and historical return rates, you can get a good idea which is usually better.
The most important thing you must account for is the time value of money. Typically you can get maybe a percent or two above inflation guaranteed return or maybe 5% above inflation return long-term through the stock market. We can therefore calculate what your returns would be if you invested all your winnings with each method and each investment strategy. Since the annuity payment is not adjusted for inflation, it's easier if we do the math ignoring inflation. All four options will be accurate and comparable, but you will have to divide by 1.8 to get the value in current dollars.
Guaranteed Returns – 3%
Annuity
If you win $1,000,000,000 and take the annual $33,333,333 payments each year, you can earn a further $45,218,850 of interest on the first payment over the remaining 29 years, $42,930,923 on the second payment, and so on. This will eventually total $585,847,190 in interest in addition to your principal, for a grand total of $1,585,847,190.
Lump Sum
If you win $1 billion dollars and take the lump sum payment, you would get $524,370,000 up front. You can then earn interest on a much larger amount of principal for all 30 years. This would give you, to begin, $15,731,100 in interest the first year, and it would then compound from then on. Totalled, this option would give you $1,235,712,254.
Stock Returns – 7%
Annuity
If you win $1,000,000,000 and take the annual $33,333,333 payments each year, you theoretically would earn on average a further $203,808,568 of return on the first payment over the remaining 29 years, $188,294,612 on the second payment, and so on. This will eventually total $2,148,692,877 in investment gain in addition to your principal, for a grand total of $3,148,692,877.
Lump Sum
If you win $1 billion dollars and take the lump sum payment, you would again get $524,370,000 up front. You can then again invest a much larger amount of principal for all 30 years. At 5%, this option would give you a total of $3,730,502,969 after 30 years.
Analysis
As you can see, if you plan on playing it safe and want to guarantee you'll be rich, the annuity will end up with more money. If you rely on riskier investments with higher rates of return, the lump sum could come out ahead. The lump sum actually becomes more worthwhile at about a 5.25% return. Depending on when you withdraw it, maybe not. You pay the price for that in the risk. Lottery winners are likely to be more okay with risk I would guess, but betting $2 and $200 million should have a emotional difference.
Another factor which this ignores is tax. Tax brackets and rates will depend on your state, charitableness, and other income; but the annuity will likely win on that front.