If you think way back to grade school, you may remember that you studied addition and subtraction one year, and then a grade or two later studied multiplication and division. Each pair of arithmetic operators intrinsically go together – they’re opposites of each other (algebraists would say inverses). What I want to explore today is that each pair of arithmetic operators carries with it a means of thinking about numbers, and thus by extension thinking about money. These two means of thinking are both useful but they’re surprisingly distinct, and people frequently make the mistake of applying the wrong type of thinking in a given circumstance. My whole premise here may seem irredeemably nerdy, and it is. But bear with me anyways – I reckon there’s real insight to be had.

Additive thinking about money is everywhere. It’s associated with he arithmetic operators addition, subtraction, and subtraction’s close relative comparison. Every bit of frugality advice clogging the internet is additive in nature – when you skip buying that $4 cup of coffee, you’ve *added* (or declined to *subtract*) $4 from your wallet. Our putative frugalist might remind you that over time these little decisions *add* up. Which of course is true. But you could twist that idea around and say the only thing those skipped cups of coffee can do is add up. I’ll bet you’ve never heard anyone say that… we’ll revisit it momentarily.

Additive money crops up in other places – when people discuss their incomes, or *compare* them, they’re really looking at the amount of money they *added* to their stash this year. The same could be said for discussions of paying off your mortgage – how much did you *subtract* from the loan principle? Saving, spending, earning, paying off – it’s all additive. You might say the financial blogsphere is painfully additive in approach.

One of the interesting features of additive finances is that events are independent. If you spend $4 on a latte today and $2 on a pack of gum tomorrow, you’re down $6. The two decisions have no impact on each other – if you didn’t buy the latte, the gum would still cost $2. For a lot of financial situations, this sort of independence is exactly the case so thinking about them in an additive way is very effective.

Of course there are lots of financial matters that don’t have this property. For starters, there’s anything with an interest rate. If you put X dollars in an interest bearing account at 2% simple annual interest, after the first year you’ll have (1.02 * X) dollars. But after the 2nd year, you’ll have (1.0404 * X) dollars. The two years are not independent – at the start of year 2 you have more money in your account than you did at the start of year 1 because of what happened in year 1 – the events are linked. This idea is frequently taught as the “magic” of compounding interest, which when you get right down it isn’t necessarily all that magical as XKCD reminds us. But this is unquestionably a different kind of thinking about money – in order to calculate how much money you’d have at the end of year 2, I had to multiply 1.02 * 1.02. We’ve left the realm of additive money and entered the realm of multiplicative money.

You can find multiplicative money in places much more intriguing than interest rates – think about job hopping and salary. If I hop from job A to job B and negotiate a 20% raise, and then three years later do it again, I’m now making 1.44x what I was before (ignoring other possible raises). Salary negotiations are multiplicative, and as a result of some good career moves I made early on I’m going to make much more, probably for life. In contrast, getting laid off and subsequent unemployment is like multiplying by a factor less than one – if you have to take a job that pays 75% as much as you used to make to get back to being employed, that likewise could easily follow you for life – multiplied into each future salary negotiation.

I mentioned before that people frequently apply the wrong type of thinking. This can go both directions, but in the salary case it’s easy to fall into additive thinking when you should be thinking multiplicative. Consider this half-assed attempt at teaching people how to negotiate salary. I’ve got news for you – worrying about the job’s expense account isn’t going to matter much. Worrying about your hair dye job is just vapid. What’s going to matter when it comes down to hardball negotiation is your walk-away number, and that’s a function of your old salary multiplied by the minimum benefit you’re willing to realize due to the job hop. You should be thinking multiplicative, and those tips are at best additive.

Now, as the XKCD comic above reminds us, repeated multiplication doesn’t always produce dramatic results. Getting big things to happen is dependent on four inputs:

- Where you start
- What the multiplication factor is
- How many times you multiply
- What inflation is

This explains why the XKCD result is so banal – the starting point is low ($1000), the multiplication factor is tiny (1.02) and the number of multiplications is pretty low (10). Nothing exciting is going to happen. When you factor in inflation over 10 years (inflation is currently about 2%) the results are going to be so close to break even as to be irrelevant.

Now contrast that with the salary negotiations. The starting point is much higher (10s of thousands of dollars per year of salary) and the multiplication factor is much bigger (1.2). So instead it takes about 4 3-year periods to double your salary – 12 years vs. XKCD’s 10. It would take about 15 year to double the salary if you factor in today’s inflation. But at the end the results are really something worth talking about. Life changing even.

It’s also possible to make mistakes the opposite direction. I see people do this all the time when dealing with savings accounts. They go out of their way to sign up for an account with a comparatively high rate (Ally is offering 0.84% right now vs. 0.10% at my credit union) and put say $2000 in the account. Guess what – this brilliant move is going to make you a whopping $17 per year. If you waste even one hour dicking around getting the account set up, there goes your profit for the next few years. Multiplicative thinking is completely impotent here – you’d make a bigger dent in your finances by not buying lattes. We can begin to see the general rule: small money and small rates favor additive thinking. Big money and/or big rates favor multiplicative thinking.

This means multiplicative thinking matters under two conditions:

- when you’re so deep in debt the rate you’re paying on your debt determines your financial outcome
- when you’re so wealthy the return on capital you’re getting on your wealth determines your financial outcome.

In the middle zone between those two, additive thinking rules. Incidentally, this goes a long ways towards explaining why the rich seem “out of touch” with middle class concerns. It’s because those concerns are additive and thus no longer meaningfully affect their outcomes. The recent faux outrage over Romney’s speaking fees is a prime example – he stated about $300K of speaking fees was “not very much”. This seems absurd from an additive perspective. Most people’s finances would be radically altered by a one-time $300K jolt. But it’s a simple statement of fact given Romney’s financial position – with something like $50M of working capital at his disposal (the exact number is private) those speakers fees represent less than a 1% change in his situation. They’re even less relevant than XKCD’s 2% interest. It’s not very much. Similarly, Romney could never change his financial situation via altering his coffee buying habits. Whether he buys no coffee whatsoever or all the overpriced lattes his bladder can hold, it won’t matter at all. All those lattes can do is add up – they can’t multiply.

As you might guess, the high finance world is much more interested in multiplicative thinking than with additive. This makes perfect sense given the huge quantities of capital involved. Trading results are also inherently multiplicative – money made trading today can be used to take bigger positions tomorrow. This explains why being active in the markets with limited capital is a fools errand, or training at best. Your trading results on $2000 of capital aren’t going to mean anything. They may help you learn to trade, but at that level of capital they will be dominated by additive effects elsewhere in your life.

Try this experiment: whenever you encounter financial advice, ask yourself if it’s additive or multiplicative. And ask whether the problem supposedly being solved is best solved with that type of thinking.