The most expensive restaurant meal in the Universe.
Last year a combination of doubt and curiosity led me to investigate a plot line in the movie “Mary Poppins Returns.” A story line which resolved the movie’s central drama claimed that the ‘tuppence’ that the original movie’s young Michael was forced to deposit in his father’s bank [rather than spending it on bird feed] had — by virtue of compound interest — multiplied to such an extent that the grown up Michael in the 2019 version of the tale was able to repay a mortgage with it.
Having recently undertaken investment training, I was able to apply my new skills to work out what two cents [standing in for ‘tuppence’] would be worth 25 years later. By assuming an average interest rate of 5% for the period [using Bank of England data], I calculated that Michael’s investment was only worth about 6.77 cents in the 1930s when the sequel was set. Not likely to be sufficient to pay off any mortgage.
This story line came to mind again recently while re-reading a favourite book, “The Hitchhiker’s Guide to the Galaxy”. Compound interest also featured in the second volume of the ‘five-part trilogy’ written by the late Douglas Adams. Fans will recall that on one of their adventures, the intergalactic travellers Arthur Dent, Ford Prefect, Zaphod Beeblebrox and Trillian [later joined by the “Paranoid Android’ Marvin], visited the ‘Restaurant at the End of the Universe’.
This restaurant is situated at the end of the universe in time. It is located in a ‘time bubble’ disconnected to the space/time continuum around it. This is convenient, since the restaurant patrons, in addition to their dining experience, have a panoramic view of the final and spectacular collapse of the universe into nothingness. Guests could enjoy their meals, including the desert trolley, and have a few drinks before returning to their own time. As the Guide itself notes, “this is, of course, impossible.”
One of the advantages of this restaurant was that there was no need to book in advance since a booking could always be made retrospectively when the diners returned to their own era. This, too, “of course, is impossible.”
You can imagine that a restaurant in such a location and at that point in time is likely to be an expensive dining option [just imagine the energy costs alone involved in maintaining the restaurant in place]. Indeed, the Guide describes the cost of a meal there as ‘fabulous’. Happily, however, this is not a problem. All that is required, the Guide notes, is that each diner — on their return to their own epoch — must deposit ‘one penny’ into a savings account and by the end of time compound interest on this very modest investment would have grown it into a sum sufficient to meet the fabulous cost of this exceptional meal experience.
My mind got to wondering again — how much would a penny invested in 2020 be worth by the end of time? Would it be enough to meet the presumably astronomical cost of this meal? ‘Astronomical’ seems to be the appropriate descriptor in this context [and I am surprised that Douglas Adams did not use this adjective!]
Investment calculation says that if we know the amount of the initial investment [Present Value], the rate of interest and the term of the investment [NPER — or number of interest periods], we can work out the Future Value [FV] of an investment.
Let us call PV one cent rather than one penny. For the interest rate, I have chosen to use the Australian Reserve Bank’s current official rate of just 0.25% — this is a historically low rate which hopefully will not last for all eternity, but who knows? At least it is not likely to fall further. [One complication may be that in certain countries which currently have negative interest rates, our ‘end of the universe’ diners may end up with an astronomical bill rather than a huge sum to pay for their meal. They may be washing dishes when they arrive at the end of time, but, then again, not for long!]
The main problem with calculating Future Value in this case is determining the NPER value. Just how long will the Universe last? In the book, the Maître D’ of the restaurant explains that at the moment of its demise, the universe has been in existence for 170,000 million billion years. He may be exaggerating as such show biz spruikers often do. Of course, this does not take into account the time elapsed since the Big Bang, but since this is estimated to be only about 14 billion years, it may not make much difference.
Douglas Adams wrote the Guide in 1978 and there are now perhaps more reliable estimates of the likely age of the universe than were available to him at the time. However, these also vary wildly. Recently, the American National Geographic Association published an estimate predicting that our Universe [there may be others] would end in a much more modest period of just five billion years. Since our Sun is predicted to extinguish itself at about the same time, I have decided to use this estimate since gourmet trail travellers from Earth at least will have to be back home before our Solar System becomes cold and uninhabitable. [1]
So NPER = 5 billion years and thus the investment equation is =FV [rate of interest, NPER, PV] or 0.25% X 5 billion X -0.01 [present value must be expressed as a negative]. So, what is the answer? A little more than young Michael’s ‘tuppence’ added up to, we hope. Let us feed these numbers into MS Excel.
To build the drama of the outcome of this equation, let us look first at how this investment grows over shorter periods of time. In this exceptionally low interest rate period, our small sum of one cent is still only worth $1.13 after the first hundred years. Just as well we still have nearly 5 billion more years to accrue interest. After 1000 years our cent is still only worth $44.58 cents which might feed a small family at the new ‘fly though’ Maccas near Mars on the way to the end of the Universe but not much more.
However, by 10,000 years our penny’s worth of investment is starting to bear real dividends. It is now worth no less than $279,172,678,585.45. That is nearly 280,000 million dollars. Not bad, eh, and that sounds like it could buy a pretty decent meal anywhere in the Universe, including the round trip on Space-X!
It is still a fair way off 5 billion years, though. However, at a mere 100,000 years, Excel is insisting on displaying the results in mathematical notion form: $1.097E + 109. That is, if I understand it correctly, 1.097 times 10 to the power of 109. That is a lot of zeross. I am not sure how to express it in words. At 281,000 years [still a long way short of 5 billion], Excel says the answer is $2.0569E + 305 [nearly 200 more zeross]. And you thought Australia’s national debt was big!
After 281 thousand years, Excel just gives up and refuses to display a number. The ‘NUM#’ error message suggests that one reason is that the number may be too big for Excel to calculate. And you thought Microsoft was all-powerful! The Excel ‘Formula Help’ function is more informative. It helpfully displays the answer as “$Infinity”. No bothering with a string of zeros or mathematical notations. Infinity! Now, that is a big number — although exactly how big is a little uncertain.
Two hundred and eighty-two thousand years is a long time, but still a long way short of five billion years. Can our penny continue to grow after this time? Infinity plus? If not, who is pocketing the uncredited interest? The bank?
In any case, I think we can now safely conclude that our diners, having an infinite amount of dollars in their ‘Restaurant At the End of the Universe’ account by the end of time should be able to pay for their meal, drinks and entertainment, even allowing for inflation!
The only thing that worries me is that if two diners turn up at the same time, both with an Infinite amount of dollars in their account, can the cashier add these two amounts up? $Infinity + $Infinity = ??? I think this sum is beyond Excel as well.
In memory of Douglas Adams. The greatest comic writer in the Universe.
Keith Harvey ©
[2] The Excel Formula help function assumes monthly interest payments.
