# A Loonie Saved

A Canadian's random thoughts on personal finance

## Aug 31, 2008

### Dogbert and log points

(I promise, this is my last math-nerd post on log points.)

Last week's Dilbert cartoon, though facetious, essentially captures how I intend to grow my money for retirement. It also helps illustrate how log points can help you compute compounding in your head.

In the cartoon, Dogbert wants people to multiply their money by a factor 10,000. Because of the long timeframe, compounding will dominate his calculations, so he can't simply say "1,000,000%/5% = 200,000 years" because he will be very far off. However, he can still do the calculation in his head if he uses log points.

We start with three rules of thumb that need to be memorized. Think of these as the "Rule of 72" on steroids:
• Rule 1: a 1% increase is about 1 log point
• Rule 2: a doubling is about 70 log points (which should be reminiscent of the Rule of 72)
• Rule 3: a 20× increase is about 300 log points
Adding log points multiplies the gains; so, for example, a gain of 370 log points equals 300+70, which gets you a 20× increase followed by a 2× increase, giving a 40× increase overall. Likewise, 230 log points equals 300-70, which gets you a 20× increase followed by a 2× decrease, giving a 10× increase overall.

If 10× is 230 log points, then 10,000 (which is 10×10×10×10) means 230+230+230+230 = 920 points. If our investment gains 5% per year (which is about 5 log points), then we need 920/5 = 184 years. This is a much better guess than 200,000 years, since the right answer is 188.7 years.

(Our estimates are still off by a bit, because all of the rules above are rounded, so each one you apply can introduce a 1% error. We applied four of them here (four 230s) so we should expect a total error of up to 4%. Rule 2 introduces the most error: rather than 70 log points, the correct value would be about 69.3, which I think you'll agree is a much more awkward number.)

You don't have to get all 920 log points from interest though: you could also get some of them by increasing your initial investment. If you start with 10× as much, you get the first 230 log points immediately, leaving you to earn the other 690 of them via interest, which would take about 690/5 = 138 years (which is close to the actual 141.5 years).

However, I have no plan to invest \$100 at the beginning and leave it for 184 years. I'm much more likely to invest every month until I retire. How do we compute this with log points?

Suppose I invest \$1/month for 10 years. Some of those \$1 installments would be invested for the whole 10 years, while others would be invested in the last month or two and would spend very little time invested; overall, each dollar spends an average of 5 years invested.

This gives us our fourth rule of thumb (which, though unrelated to log points, is still very useful in log point calculations):
• Rule 4: Periodic equal payments at equal interest are (practically) equivalent to one lump sum invested at the same rate for half as long
So, applying this rule to the Dogbert scenario: instead of investing \$100 for 184 years, I could divide that \$100 into equal (tiny) payments over twice as long (368 years) and still end up with a million.

Thanks to these four rules, I did these calculations entirely in my head, using a calculator only to confirm my results.

## Aug 29, 2008

### Log points

Ok, you may all think I'm crazy for this one, but I find the easiest way to get a quick picture of the performance of my investments is through the formula 100×ln(x), a metric I refer to as log points (though its technical name is the rather unfortunate "centinepers"). "Log points" columns crop up in most of my financial spreadsheets.

Why on Earth would I use this formula? It makes the spreadsheet do the hard math, so that whenever I want to compare two numbers, I only need to do simple mental subtraction of relatively small quantities.

Suppose you have an asset worth \$36316 one day and \$35596 the next day. Quick: what percentage did you lose? Well, you need to subtract them, then divide by an unwieldy 5-digit number. That's ok if you program your spreadsheet to do it, but if you have a large amount of financial numbers and you want to compare them all to get the gist of them, you'll have a lot of pairwise formulas to enter into your spreadsheet. Instead, suppose we convert these two numbers to log points, using the above formula. So what?

Here's the trick: the difference between two log-point amounts equals the percentage difference between the original two dollar amounts. This works whenever the difference is small; anything less than about 10% works just fine. In our example, the first number comes out to 1050, and the second is 1048, so we've lost 2 log points, which indicates that we've lost very nearly 2%.

If we let the spreadsheet compute each number's log-point equivalent, we can pick any pair of dollar quantities and compare the percentage difference just by mental subtraction. This greatly aids visual scanning to find patterns.

Log points have a number of other interesting properties with regards to compounding and amortization that I may describe in a future article if anyone's interested.

(Nobody was interested, but I wrote another article anyway. Part 2 is here.)

## Aug 24, 2008

### Reducing gas price volatility

When a gas station increases the price at the pump, they should be required to sell it at the new higher price for, say, seven days before lowering it again.

With this rule, a gas station that increases its price too much too soon will get stuck being the only one selling at that high price for a whole week, during which drivers will buy their gas elsewhere. Every time this happens, they may lose a week's worth of revenue, which is about 2% of their annual revenue.

Gas stations will be forced to absorb price fluctuations to keep their prices competitive. They'll have to keep inventories higher so that they can ride out whatever supply situations they normally use to justify raising prices. They may need a cash reserve on hand in case they do get stuck with lower revenues for a week. Existing competition laws will make it illegal for different companies to agree to keep their prices high and nullify this effect.

What do you think?

## Aug 2, 2008

### Descriptive budgeting

Budgeting can be a real downer if you've never done it before, since it seems to limit your freedom to do what you want with your own money.

However, let me point out an important budgeting fact: everyone has a budget; it's just that many people don't know what theirs is.

If you've ever wondered why your income doesn't afford you more savings or a better standard of living, the first step in budgeting is to look at it as descriptive rather than prescriptive. What I mean is: don't start by designing a budget to control your expenditures; rather, simply write down what your expenses are.

In the first draft, it doesn't even have to be that accurate; just include the expenses you can think of off the top of your head: housing, groceries, utilities, entertainment, transportation, etc. Then, for a couple of months, make your purchases by debit card, or (a bit more dangerous for those who aren't good with money) credit card, so you can track your expenses and see whether they match your prediction. Adjust your prediction until you get to a precision that satisfies you, call the remainder "petty cash", and now you've got a good picture of your cash flow.

The bigger the petty cash account, the easier it is to write the budget, but the less precise it is. If you're ok with not knowing where, say, 20% of your income goes, that's fine. Hey, it's your budget.

Once you have a budget that matches your actual situation, you can look at it and decide whether you are getting the results you want from your hard-earned money. Only if you are unsatisfied with your budget do you need to modify it and change your spending habits, making your budget increasingly prescriptive until you like what you see. Start with the biggest expenditures, since changes there will have the largest impact on your bottom line. You can refine your budget as you go until you get the effect you want.

And take it easy: unless your debt is spiraling out of control, there's no big rush. For our family, we took a leisurely 7 years or so to refine our initial descriptive budget into our current budget. We tweaked it every time we felt we weren't getting the results we wanted. Encouraged by watching our savings grow, we've now got quite a detailed budget, with just 3% of out income going to petty cash, and the rest accounted for.