With the Mega Millions seemingly on fire at over $790 million it is worthwhile looking at some of
the probability math of this game. We noted previously that
one could purchase all 302 million number combinations for $605 million [or $2
a ticket]. Afterall, what else would be your
odds of winning a proftable prize?
Let’s initially
refamiliarize ourselves with the mathematical formulae for an annuity-due, 30-years at constant value
payment. We’ll assume a $660 million
jackpot with interest rate [i] of 2.4%:
Where v=1/[1+i]=0.977,
and d=1-v=0.023. Or a X=660m/21.7, and an
annual annuity of $30m [versus $21m if a typical deferred, immediate annuity].
The issue is on
one hand the lottery has continued to keep i artificially low at 1.4%.
So for the same $30m annuity stream we
have v=1/[1+i]=0.986, and d=1-v=0.014.
Or a 24.7 times X=$30m, or a rise↑ to $741m based only
on their bogus, assumed interest rate! Closer
perhaps to where the lottery in fact rose between recent drawings [from ~$645
million, to ~$770 million.]
Jointly, the lottery can also provide a growth estimate [k] of the annuity
instead of keeping it at a stable value.
In theory the k should be less than i, else
the valuation model would explode.
Instead the lottery chose a much higher k than even pro-coronavirus
2.4% or current 3.4%, etc. They have 5%.
Which when k=0%, collapses back to the result further above.
And when k=i=2.4%, the formulae fails but the result is still an
intuitive $660 million/30, or an annuity dropping to a $22 million initial
payment. Based solely on the lottery’s
selected high value for k.
But what happens when k is 5% where the lottery
currently has it set? The same $600
million jackpot would only need an initial payment of $15 million. In other words, the cash lump option would -all
things being equal- be reduced↓ by -32% [$22 million,
to $15 million] solely because of the fictitiously higher, assumed growth rate.
And this lowering of i relative to k
has allowed this week’s jackpot [top 3] to rise much faster than the cash payment value [top 7] on the annuity option. See this 5-year historical chart
below with all of the largest jackpot rankings on both prize statistics.
We see the impact of the actuarial annuity calculation above to
tinkering of the lottery assumptions on your game value. With stated jackpots rising above average, while
[your more critical] cash lump options decreasing. And coming full circle with the strategy of buying all the tickets, this cash value would have to be
considered. Still, we see this week as
the likely apex of this lottery jackpot run, with over a ¾ chance of a win
later this July 2022.
supporting salil statistics:
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