Win-a-Million

I was wondering if I was to offer a 'million dollar giveaway' of a dollar a year for a million years, what would the current lump sum payout value be?

How about a dollar a day for a million days?

In both cases, the only practical meaning of a 'lump sum value' is whatever an arm's-length investor would be willing to pay to receive this return. This in turn would depend on their estimates of future inflation rates and of how well you and your successors can be depended on to keep up the payments.

In that case I would have to say the value would be extremely small. Suppose you won the million dollars, and I was considering paying you a lump sum so that it is paid to me. As I am now 43, this prize payout would only be worth probably at most 70 or so dollars when I die. Even if I wanted to invest the money 'wisely' so it would be worth more than that, I would be in my 90's before it would be worth $50, which is probably the minimum amount to make any kind of meaningful investment, so it would not be worth it for that.

If I purchased it for my grandchildren, or their children, or their children, it would take so long to grow to any size (assuming that a generation is 20 years, it would take 50 generations to grow to $1000, which does not even pay for a semester at many colleges, assuming there are no tuition hikes in the next 1000 years) that it would not be worth it for that.

If I were considering purchasing it for a business, it would take several hundred, if not several thousand years (even if invested 'wisely' for the amount to have any meaning to a business, and that is assuming there is no inflation, that your business is still around in several hundred to several thousand years, and probably a bunch of other things I am not even thinking of.

With those things in consideration, if you won the prize, and offered to 'sell' it to me for a lump sum, I probably would not even consider it unless your asking price were less than $10, and I might not buy it if it were less than $1, such as 50 cents. I suspect that companies that do this have a formula, but it would be hard for me to imagine that under those conditions ($1 a year for a million years) it would be very much.

Brian Christiansen

So this being puzzleland, we make the necessary assumptions: 1. The value of money is 10% per year 2. The successors can be relied on 100% to keep up payments.

Without doing the math precisely, we can get a pretty good idea:

It can be seen that any investment that accrues more than $1 per year from day 1 is always better than the million dollar giveaway, since the giveaway can never catch up.

Thus it is worth around $10.

In fact the lower the rate of return, the more it is worth: At 1%, it's worth about $100.

At zero return, it's worth $1000 000.

The daily question is the same, but with a rate of return 1/365 of the yearly one, so it's worth around $3650 if money is worth 10%pa.

Jim Ward schrieb:

It doesn't really depend on being a million; just go to a bank and find an investment that pays a dollar interest per year. You can recoup your investment after a million years, so you could probably slice something off the investment to have it used up after that time, but the slice would be marginal at best.

This is slightly more difficult, as interest isusually not due daily, but it shouldn't be too hard to figure out something that stays in equilibrium (i.e. infinite payout of a dollar a day).

Cheers

Patrick Hamlyn schrieb:

After the first year, your account looks like this:

Capital:

3650.00+ investment 365.00- payout

If we assume, say, 5% interest, I guess we want something like

f(n) = 1.05 f(n-1) - 1, f(0) = a, f(1e6) = 0

This is an easy recurrence to solve. Let g(n) = f(n) - 20, so

g(n) = 1.05 g(n-1) = (a-20) * 1.05^n

so

f(n) = (a-20) * 1.05^n + 20

a = 20(1-1/1.05^(1e6))

So, as I see it, that makes a difference somewhere past the 20000th decimal place....

$5,000 is 2.5% of $200,000. Perhaps you could put that amount aside, then hire someone to manage the annual dispensation and ongoing maintenance for 2.5% of the principal forever?

Cash Value 0.01c. Void where prohibited. (thats what it says on the back of my ticket).

Th general formula for a perpetuity, a fixed payment for an infinite period of time is (Payment Amount)/(Interest Rate). A million payments is close enough to infinite for this formula to be awfully close.

To answer the second question, you'd have to calculate the effective interest rate for one day, and use the same formula.

A good estimate for the discount rate on either one would be a long-term bond. If you are as reliable as the US government, use the rate on the longest outstanding Treasury bonds, which is 4.79% today. I'll use 5% for simplicity.

At a 5% interest rate, a bond which pays a dollar a year for a million years is worth $20. If you could re-invest your dollar in similar bonds forever, you would preserve the income stream.

A bond which pays a dollar a day for a million days pays $365 a year for about 2700 years. If it paid $365 at the end of each year, it would be worth $7300, but since each year's payments are spread evenly over the year, you would gain an average of 2.5% on the reinvestment of each year's payments during the year, raising the value to about $7500.