Lottery Winner Rejects $1 Million For $1000 Weekly Payments For Life – Mind Your Decisions

Here’s a story that has captured the attention of the internet. A Canadian woman in her 20s won the grand prize of a lottery. Instead of taking the $1 million lump sum payment, she opted for $1000 weekly payments for life.

Many people were questioning the choice financially. And this is actually a perfect situation to analyze a real world math problem!

How much is a lottery payment of $1000 weekly for life worth to a 20 year old in Canada? Let’s do the math!

As usual, watch the video for a solution.

Math Just Got Important

Or keep reading.
.
.

“All will be well if you use your mind for your decisions, and mind only your decisions.” It costs thousands of dollars to run a website and your support matters. If you like the posts and videos, please consider a monthly pledge on Patreon.
You may also consider a one-time donation to support my work.

.
.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Lottery Winner Math

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Which is monetarily worth more: $1 million today, or $1000 weekly for life? We need to make a few assumptions to do the calculation. But this particular problem has many special details that make it perfect for mathematical analysis.

First of all, there is no tax on lottery winnings in Canada. So we don’t have to worry about the lump sum payment getting hit with higher income tax than the weekly payments.

Second, for weekly payments over many years, you might be worried the company goes insolvent and does not pay. But Loto Quebec is a crown corporation, so this lottery payout has a very low risk of not being paid.

The $1000 per week prize is also guaranteed for some number of years. In this case, they will guarantee 20 years of payments to the winner or a designated heir. So we know the lottery winner will get a nominal payout of:

(20 years) x (52 weeks/year) x 1000 = 1,040,000

So in either case a winner would get $1 million nominally paid. (A winner above 71 has a reduced schedule, but our winner was in her 20s so we will ignore this detail).

Furthermore, a 20 year old may expect to live another 65 years. So the payout can be:

(65 years) x (52 weeks/year) x 1000 = 3,380,000

Perhaps getting 3 times as much money, spread out over your life is not so bad?

There is also the non-financial aspect that lump sum winners often blow their money irresponsibly, or they become targets of requests for help due to their newfound wealth, so an individual may not want to have the lump sum payout be known publicly.

But we can’t merely compare $3 million over 65 years to $1 million paid immediately–that is apples to oranges. We need to compare the present value of the $3 million payout so the lump sum.

Time value of money

Money in the future should be worth more than money today, and it should grow by a specified interest rate.

Let’s say you can guarantee earning i = 5%. Then we can consider the future value of $100 over 2 years by multiplying by the growth factor (1 + 5%):

year 0
$100

year 1
$100(1 + 5%)
$105

year 2
$100(1 + 5%)²
$110.25

We can similarly reverse the calculations to take a future value and “discount” it by the annual discount rate of 1/(1 + 5%) to get a present value.

year 2
$110.25

year 1
$110.25/(1 + 5%)
$105

year 0
$105/(1 + 5%)
$100

Valuing the annuity

Let’s consider a general annuity that has a payment of R at regular intervals, and we have an periodic interest rate of i. Suppose we are getting payments in periods 1, 2, …, T. Then our stream of payments will be:

period 1
R

period 2
R

…

period T
R

We can discount each payment by the corresponding time frame to calculate the present value of each payment:

period 1
R
R/(1 + i)

period 2
R
R/(1 + i)²

…

period T
R
R/(1 + i)^T

Summing all these present values will yield a net present value, or a value for the annuity. We wish to sum:

R/(1 + i) + R/(1 + i)² + … + R/(1 + i)^T

This looks challenging to sum, but it can be worked out by a well-known formula. The sum of payments is a geometric sequence with a first term R/(1 + i) and a common ratio of 1/(1 + i). By using the sum of a geometric series we have the resulting formula:

[R/(1 + i)][(1 – (1 + i)^-T)/(1 – (1 + i)^-1)
= [R/(i)][1 – (1 + i)^-T]

In the Canadian lottery example, we just need to input values relevant to the situation. The weekly payments are R = 1000. Assuming 65 years of payments, that is 65×52 = 3380 payments.

We do need to make an assumption about the interest rate i, which is often an important factor in present value calculations. It is a good idea to test a few values to see how sensitive the result will be. Let’s start with an annual interest rate of 5%, so we have a weekly interest rate of i = 5%/52.

Substituting those values into the formula gives:

(1000/(5%/52))[1 – (1 + 5%/52)^-3380]
≈ 999,612

So we can see the present value of 65 years of $1000 payments is nearly the same at the $1,000,000 lump sum, if we take an interest rate of 5%.

But if we raise the interest rate to 8%, as stocks have yielded 8 to 10% in the long term, then the present value will be much lower:

(1000/(8%/52))[1 – (1 + 8%/52)^-3380]
≈ 646,400

Revisiting the decision

Would you take $1 million as a lump sum or $1000 payments for life? I think most of us would take the $1 million, as it has a potential to be more money. However, lump sum winners are often the targets of people asking for money, and there is a risk that some bad investments will ruin their return. So it’s not a simple open and shut case for me to reject the weekly payments of $1000.

Another application: Bobby Bonilla’s contract

The above example shows how a lump sum can work out better. So I want to give a flip side a famous sports contract in which annuity payments would be favored by most.

On July 1, from 2011 to 2035, the retired baseball player Bobby Bonilla gets paid about $1.19 million from the NY Mets. That’s right! A retired player still gets more than $1 million annually, and a total payments of about $30 million.

How did this come to be?

In 2000, the active player Bonilla had $5.9 million remaining on his contract. His agent Dennis Gilbert, who was also a life insurance agent, proposed converting the contract into deferred annuity payments. He calculated delayed from 2011 to 2035, for some fixed amount R, at an annual interest rate of 8%. How much would the annual payment have to be? Our payments will be:

year 2011 = 2000 + 11
R
R/(1 + i)¹¹

year 2012 = 2000 + 12
R
R/(1 + i)¹²

…

year 2035 = 2000 + 35
R
R/(1 + i)³⁵

We wish to find the net present value:

R/(1 + i)¹¹ + R/(1 + i)¹² + … + R/(1 + i)³⁵

We have 25 payments, with a first term of R/(1 + i)¹¹, so the geometries series has a sum of:

[R/(1 + i)¹¹][(1 – (1 + i)^-25)/(1 – (1 + i)^-1)

We need this value to be equal to the $5,900,000 remaining on the contract in 2000. And we will use an interest rate of i = 8%. So we get:

[R/(1.08)¹¹][(1 – (1.08)^-25)/(1 – (1.08)^-1) = 5900000

We can solve this equation for R, to the nearest cent, to get:

R = 1,193,248.20

And this is the exact amount, to the cent that Bobby Bonilla is paid every year!

So the concept of net present value is used even in multi-million dollar contract negotiations! Yes, math does have real world applications!

Why did the Mets agree?

At the time the Mets were getting 15% returns from Bernie Madoff, so deferring payments at an 8% interest rate seemed like a bargain. They could invest the remaining amount on the contract and make a lot more than the annuity payments.

However, Bernie Madoff was exposed in 2008 as the largest known Ponzi scheme. At a time the economy was taking a downturn, Bobby Bonilla was getting guaranteed million dollar payments and that captivated the internet.

So July 1, every year from 2011 to 2035, is known as “Bobby Bonilla” day, a day to celebrate a famous sports contract where the agent used math to guarantee appealing payments to his client.

References

https://www.facebook.com/realratedred/posts/a-20-year-old-lottery-winner-surprised-everyone-by-turning-down-a-1-million-lump/1237089611787105/
https://www.reddit.com/r/theydidthemath/comments/1pjakjl/self_how_is_that_a_better_deal_and_why_would_the/
https://www.reddit.com/r/interestingasfuck/comments/1pj9bst/20yearold_lottery_winner_turns_down_1m_cash_for/
https://ca.news.yahoo.com/20-year-old-lottery-winner-decides-against-1m-lump-sum-opts-for-lifetime-weekly-annuity-in-hopes-of-buying-a-home-120026248.html
https://www.ctvnews.ca/montreal/article/montreal-woman-wins-1000-a-week-for-life/
https://en.wikipedia.org/wiki/Loto-Qu%C3%A9bec
https://www.mtlblog.com/montreal-lottery-winner-1000-for-life
https://loteries.lotoquebec.com/en/winners/stories?pagesGaleries=communiques-de-presse:5&histoireuuid=806ae605-d112-4801-8e69-5b2f1a60458b#maincontent
https://assets.lotoquebec.com/ressources/assets/v3/assets/blt8296e79a7001648c/blt6561fe89ebf15b0b/6841ac48df13920afdf5ea82/gagnant-a-vie_rules_en.pdf
https://www.bankofcanada.ca/rates/related/inflation-calculator/
https://en.wikipedia.org/wiki/Annuity
https://en.wikipedia.org/wiki/Time_value_of_money
https://www.espn.com/mlb/story/_/id/16650867/why-mets-pay-bobby-bonilla-119-million-today-every-july-1-2035
https://www.letsassemble.com/blog/the-bobby-bonilla-retirement-plan
https://dennisjgilbert.com/dennis-gilbert-contracts/

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2025.

(US and worldwide links)

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.2/5 stars on 564 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 81 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.2/5 stars on 55 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 148 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.5/5 stars on 57 reviews)

Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 138 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 45 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.3/5 stars on 38 reviews)

KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon’s “Kindle Unlimited” program. Included in the subscription you will get access to millions of ebooks. You don’t need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

US, list of my books (US)
UK, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
India, list of my books (IN)
Australia, book results (AU)
Italy, list of my books (IT)
Spain, list of my books (ES)
Japan, list of my books (JP)
Brazil, book results (BR)
Mexico, book results (MX)

MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.