Puzzling with Win Smith
Last month I had the pleasure of meeting Win Smith, who has a fantastically interesting career doing financial modeling. Among his other endeavors, his blog The Well-Tempered Spreadsheet occasionally presents puzzles of a financial bent. One interesting puzzle cleverly hit a financial angle, but is stated very succinctly:
Gregory was born on a Monday in New York City. His father sent out Western Union telegrams to announce the birth. The family later moved to London. On his seventh birthday, which was also a Monday, the boy saw the British monarch.
Was the monarch a king, or a queen?
I donít want to give away the answer, but you can read the comments on the original post here and here.
Two more obviously financial puzzles involve pool loan-to-value ratios:
The weighted average LTV is a standard measure of the quality of a pool of mortgages. The weighted average LTV is calculated by weighting each LTV by the respective loan amount, and then dividing the sum of the weighted LTVs by the total loan amount.
Suppose we have a $6 million pool of mortgages with a weighted average LTV of 80%. We would expect the total property value to equal $7.5 million = $6 million/.8.
But the total property value is actually $9 million.
How is this possible?
And also the present value of US Treasury Debt.
Make the following assumptions:
The total debt is exactly $11 trillion.
- All debt securities pay interest at fixed rates. Treat T-Bills and inflation-protected securities as if they make fixed interest payments on fixed principal amounts.
- When any security matures, it is refinanced into an identical security with the same principal and interest rate. The refinancings continue indefinitely.
- The discount rate for each security (and its successors) is the same as its current interest rate.
I find these to be like Zen koans. Just a fun was to ease back into work after a break.
For those of you who donít want to actually solve the puzzles but still want the answers, read on below. But SPOILER ALERTÖthe answers are coming.
1. The trick in this puzzle is to think of the dates, and date math is critical in financial calculations.
Have you ever noticed how your birthday moves one day a week forward each year? So if your birthday was on a Friday this year then it will be Saturday next year. The exception to this is in a leap year when (if youíre born in March or later, as about 10/12ths of people are) your birthday will move forward two days.
Now back to Gregory. Heís seven, so weíd expect one or two leap years in his short life, but heís never seen a February 29th since his birthday is a Monday at birth and at age seven. How can this be?
It all goes back to how we define & estimate a year. The Romans had figured out the leap day concept, declared that a year was 365.25 days and started the practice of adding a leap day every four years. The problem is that the time it takes the Earth to revolve around the sun is actually closer to 365.2425 days, which sounds pretty close, but that difference adds up. After over sixteen centuries, the calendar was misaligned with Earthís position relative to the Sun by over twelve days. The Gregorian calendar was corrected for this and declared that every 400 years there would be 97 leap days (97/400 = 0.2425). So on the century years (e.g. 1800, 1900, 2000) itís a leap year only if the first two digits are evenly divisible by four. Thus 1900 was not a leap year but 2000 was.
Now we can solve the puzzle. Gregoryís parents sent a telegram after his birth, so he was born after 1800. Thus the only period with no leap year for a seven year old would be to be born between 3/1/1896 and 2/28/1897. So Gregory was seven in late 1903/early 1904. As Queen Victoriaís 63-year reign ended in 1901, the monarch Gregory saw was King Edward VII.
2. Weighted averages are funny things. Iíve had fits in the past with the optics of how to show a weighted average calc of a year versus the months, and this is a variant. The trick is in the definition of a Weighted LTV. LTVs are Loans/Property Value, but in this case the weight isnít the percentage times the property value, but instead itís weighted by the Loan amounts. Thus as a simple example, if the pool has two properties, one worth $7.5M and one worth $1.5M. The larger property has a $6M loan on it, or an 80% LTV. The smaller property has a $1 (not $1M) loan, so itís LTV is essentially zero. The weighted average LTV is still 80%, but the portfolio is worth $9M and only has $6M of loans. Go figure. The full solution set to this is more like an ellipse, and Win Smith has a good discussion of this on his website.
3. The answer is $11T. If you get a coupon at a rate into perpetuity and discount at that rate, the result is the starting principal. That was my intuition, and Win has detailed comments with more color.
So, my fellow financial puzzlers, I hope you enjoyed reading this as much as I did writing it!
Puzzles copyright Win Smith. Reprinted with permission.