# Why compare asset performance using returns, instead of using change factors?

Why do investors compare the returns of assets as opposed to the change factors of assets? E.g. comparing `6% / 4%` instead of `106% / 104%` to get the comparative performance of two assets.

AFAIK comparing performance using `return A / return B` precludes you from:

• Comparing returns where one return is 0% - see example 1.
• Comparing returns with different signs - see example 2.

Example 1: Infinitely better/worse returns:

``````Return A = 6% (invested £100 and received £106)
Return B = 0% (invested £100 and received £100)
``````

A is infinitely better than B - `(0.06/0.00)`

Example 2: Incomparible returns:

``````Return A = 6% (invested £100 and received £106)
Return B = -2% (invested £100 and received £98)
``````

A is better than B... but by what? `(0.06/-0.02) = -3` (not meaningful)

Example 3: Exponentially better performance after inflation

``````Return A =  6%
Return B =  4%
Inflation = 3%
Return A (adj) = 2.91% = ((1 + 0.06) * (1 + 0.03)^-1) - 1
Return B (adj) = 0.97% = ((1 + 0.04) * (1 + 0.03)^-1) - 1
``````

A is 300% better than B - `(0.0291/0.0097)`

``````Return A =  6%
Return B =  4%
Inflation = 3.5%
Return A (adj) = 2.42% = ((1 + 0.06) * (1 + 0.035)^-1) - 1
Return B (adj) = 0.48% = ((1 + 0.04) * (1 + 0.035)^-1) - 1
``````

A is 500% better than B - `(0.0242 / 0.0048)`

Note: this example is not so much a problem - just an observed difference in behaviour when comparing assets by return as opposed to by change factor. When comparing by change factor, the comparative performance of the returns remains the same when a coefficient (such as inflation) is applied.

• Investor compares 6% - 4% = 2% outperformance. Not sure why you are actively dividing stuff. – base64 Dec 10 '15 at 14:43
• I've seen people such as John Bogle compare returns by dividing (not saying that makes it right/better - it's just an observation). In Bogle's particular scenario, he's stating that fund A made 57% of the profit of fund B before inflation, but that drops to 53% of the profit after inflation. He's demonstrating the effect I've outlined in example 3. He continues to state "deducting the same inflation rate from both figures further increases the comparative advantage of the investment with the higher return" – Lawrence Wagerfield Dec 10 '15 at 14:54
• And to deal with inflation, investors uses Fisher Equation. Inflation figure is the same for entire country and doesn't affect the comparison of two assets. – base64 Dec 10 '15 at 14:54
• I think he is trying to exaggerate stuff and emphasizes that one certain asset has high (real) return. – base64 Dec 10 '15 at 14:56
• Inflation affects the difference between two real return over time, but does not affect ordinality of the "best" asset. If asset A has higher nominal return than asset B and higher than cash (0%), asset A will always be preferred. Subtracting whatever common inflation percentage does not improve decision making process. – base64 Dec 10 '15 at 20:03

## 2 Answers

I think what you're describing is just a notational convenience for percentages that has no specific relationship to investment returns. Why don't cereal boxes, toothpaste tubes, etc. trying to attract consumers say "Now contains 120% of what it used to contain!" instead of "Now 20% more!"? Why don't coupons say "Pay 90% of what you would otherwise pay!" instead of "Get 10% off!"? Why don't newspaper articles say "The population of Podunk (or the price of gas or whatever) increased and is now 103% of what it was last year" instead of "The population of Podunk (or the price of gas or whatever) increased 3% last year"? Specifying changes by referring to the difference from 100% is common practice across many situations where percentages are used, and the usage in investment returns just maintains that convention.

• The marketing and vanity reasons make sense. Even charts become more intuitive when there's a 0 to partition the Y axis! However, with my current understanding, people should be wary if someone (like Bogle) starts relatively comparing returns. For example, if you believe a 10% return is 2x better than a 5% return, you're implicitly agreeing that a 5% return is infinitely better than a 0% return. Additionally, the inability to compare +/- returns with this method compounds my belief that it's not a practical way of measuring performance - which tells me its only used to exaggerate/sell. – Lawrence Wagerfield Dec 11 '15 at 17:14
• @LawrenceWagerfield: To be honest, I don't see people doing that sort of comparison all that much (that is, saying that one investment is "twice as good" as another). People do compare returns to see which is better (and, as someone commented, this works fine for ordinal comparisons), but not "how much" better. In the instances where they do try to quantify "how much" better, it seems more common to use dollar amounts calculated at some particular time interval (e.g., after 10 years this investment will earn \$1000 and this other one will earn \$2000). – BrenBarn Dec 11 '15 at 19:27

I think it makes good sense to say that an investment that increases in value by, say, 8%, is twice as good as one that increases by 4%.

Suppose I am planning to live off the income from my investments. I invest \$400,000. If potential investment A returns 8%, then I am getting \$32,000 per year. If potential investment B returns 4%, I am getting \$16,000 per year. \$32,000 is twice as much as \$16,000. It is perfectly reasonable to say that investment A returns twice as much as investment B. It would be completely wrong to say that investment A returns only 108/104=1.038 or 3.8% better than investment B.

If A returns 8% while B returns 0%, then I'm comparing \$32,000 to \$0. It is plausible to say that A is infinitely better than B. A returns something while B returns nothing.

You don't preclude the "infinity" possibility by using a growth factor. What if investment A returns 8% while investment B goes bankrupt? That is, B loses 100%, so it is now worth 0. Then A / B = 108 / 0 = infinity. (Sort of. Mathematicians have a lot to say about calculations like this.)

Sure, if investment A makes 4% while investment B loses 2%, it's pretty meaningless to say that investment A produced negative 2 times as much profit as investment B. But ... A had a profit while B had a loss. They're going in opposite directions. If two trains leave Chicago, and train A heads towards Los Angeles while train B heads toward New York, how must sooner will train A reach Los Angeles than train B? The question is almost meaningless.

• I agree with elements of what you're saying - the point about investment B losing 100% is particularly strong. The only part I disagree with is the final paragraph: investments fluctuate all the time, flipping between making gains and losses. I wouldn't say it's meaningless to compare them during a snapshot where one makes a gain and the other a loss. – Lawrence Wagerfield Dec 11 '15 at 17:22
• @LawrenceWagerfield My point was that there's no clear way to compare a positive number to a negative number as a magnitude -- "A is 3 times as good as B" -- that's really meaningful in this context. Sure, investments fluctuate all the time. A company that lost money this year might well make money next year, etc. I guess my analogy to trains heading in different directions is flawed in the sense that it gives the impression that the direction is fixed. That wasn't my intent, I was just speaking of where each was headed for the time period under discussion. – Jay Dec 14 '15 at 6:06