The Case For A Geometric Mean For Quoting Returns
Investor · Round Rock, TX · Member since 2010 · 8k+ posts · 4k+ votes
Return citations were getting thrown around on another thread a few weeks ago so I took the liberty of dusting off some old finance texts where people try to get quite specific about citing returns. The case is made frequently for using a geometric mean instead of an arithmetic mean. There are also frequently negative returns with large enough samples so people like to add one to each of the observations and subtract one from the output of the geometric mean using these observations to arrive at a return metric.
So in other words you would use:
=GEOMEAN(Range_of_Percentages + 1) - 1
This mean is always equal to or smaller than the arithmetic mean that most people use to cite returns over a period of years. Presumably this is why it is not used even though it is more accurate.
Does anyone use this when they analyze their portfolio's returns? If so, do you use the percentages annually or make it more granular? Are there any problems you see with using geometric mean instead of arithmetic mean other than the greater difficultly in arriving at the figure?
Well, in assets such as real estate, which do not have mark to market capacity, the way it is accounted for is only dealing with the actual. As value is not true until liquidation is achieved.
Measuring your net worth only shows that you increased how many assets you own, it is not a bench mark of management. Consider the following:
T1 = 10
T2 = 50
T3 = 30
So in year two, what is illustrated that you acquired more assets, 40 more to be exact. So you had a 400% increase in your worth. So what? Perhaps someone gave you $1 Million to invest with. In year three you sold assets, 20 worth. So your net worth dropped by -40%. Again so what, you liquidated some assets. It is not reflective of what gains or loss you had, merely the fluctuations in your asset schedule.
You could have gone from 10 in year one to 30 in year three with millions in loss. Adversely you could have gone with millions in gains.
Financial institutions do the net worth calculation you are talking about in relation to AUM. Which is not relative to return in any manner. You can have 200 Billion AUM and have a return of -50%. In year one, say a fund opens and perhaps it only raises 500k and then in year two say it raises 10 million. It is important to understand the raise since it might reflect in the return if some of the money was not fully invested to generate a return. However, it is a footnote to the return number and is not a major talking point in the discussion. Return is return and yield is yield.
Including cash flow is what is suppose to be done. To not include the cash flow your just doing math on numbers. Return includes cash flow by definition. Yield includes cash flow by definition. Its not complicated, it is proper.
You can do what ever you want, it doesn't mean it is right. If you mentioned to me your NW was X, I would still ask you what returns you had. If you use the difference between year beginning NW and year end NW you simply would be wrong. It has nothing to do with distortion in accounting, what you are doing is just trying to create a meaning from something that doesn't mean what you intend it to mean.
What do you mean by "more accurate?" Which one is more accurate is going to be determined by the specific question trying to be answered.
Saying geometric mean is more accurate than arithmetic mean would be like saying "multiplication is more accurate than addition." Clearly that statement doesn't make sense, and the answer is going to depend on whether you're trying to generate a product or a sum.
By the way, while you probably know this Bryan, others may not get the distinction...the reason I chose that analogy above is that arithmetic mean is an attempt to determine the answer to the question: "If all quantities had equal value, what would that specific value have to be in order to achieve the same sum?" And geometric mean is an attempt to determine the answer to the question: "If all quantities had equal value, what would that specific value have to be in order to achieve the same product?"
So, saying one is more accurate than the other REALLY IS like saying "multiplication is more accurate than addition."
All that said, when I quote basic returns, I don't generally use arithmetic or geometric mean, so maybe I'm missing your basic question -- what's the specific question you're trying to answer by using one or the other? Perhaps if you link to the other thread you're talking about, it will be clearer?
This issue is distortion J. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?"
Here is an example pulled from another math site:
//Quote
For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant quantity is the geometric mean of these three numbers.
//End Quote
The arithmetic mean in this example is 30% and the geometric mean is roughly 28%. In other scenarios the distortion can be MUCH higher.
To me it seems the geometric mean with an adjustment for negatives is much more accurate for what people are often trying to describe. I did the analysis on my portfolio over the last 7 years and the difference is only .8% (21.5% versus 20.7%). Maybe the small delta in typical scenarios is why people tend to not worry about it too much.
Sure, if you're trying to average percentage returns, geometric mean is better than arithmetic mean, for certain. Likewise, if you're trying to average dollar amount returns (for example, I earned $1000 on one deal, $3000 on a second deal and $5000 on a third deal...what's the average return on a deal?) then arithmetic mean is clearly more accurate.
Personally, for the example you used, I prefer IRR, as it's more meaningful than either of those as an answer to the real question I should be asking.
In fact I don't think I've ever asked or been asked a question where geometric mean is going to provide the the meaningful answer. And when it comes to determining success of a series of investments, I can't imagine arithmetic mean ever being very meaningful. Regardless of whether either is accurate, to me I haven't found them to be meaningful.
Personally, for the example you used, I prefer IRR, as it's more meaningful than either of those as an answer to the real question I should be asking.
Only things meaningful to me are those on my balance sheet personally.
Personally, for the example you used, I prefer IRR, as it's more meaningful than either of those as an answer to the real question I should be asking.
Only things meaningful to me are those on my balance sheet personally.
Then how do you know how much money you're making??? :)
I don't how I would use IRR given the inputs. IRR requires cash flows, which are not a given in the example cited. How would you use an IRR given these inputs?
I couldn't.
But, that was my point above -- I don't consider "average rate of return given three annual percentages" to be a meaningful answer to anything I might want to know.
I completely agree that geometric mean would give a better answer to your question than arithmetic mean, but I don't think the question/answer gives any good insight into my business' results.
And that may very well just be because I just value different metrics (in other words, I'm not making a judgement, just an observation).
Fair enough. What is another measurement that is easy to look at using historical balance sheet data given weekly? That is what I have. I simply totaled net worth gains annually, took the arithmetic average returns annually, and used those to come up with an overall geometric average. Surely this is incorrect or inaccurate.
What would you use if you have net worth gains weekly for the last 7 years?
You really shouldn't calculate it like the example suggest to begin with. This stems from mixing up yield versus return.
The example fails to convey properly when the investment concludes. Is it like Example 1 or one of the sets in Example 2 (A,B,C)
Example 1
T0 = -100
T1 = 10
T2 = 50
T3 = 30
Example 2 (A,B,C)
T0 = -100 / -100 / -100
T1 = 110 / 10 / 10
T2 = 0 / 150 / 50
T3 = 0 / 0 / 130
In Example 1, the Geometric Mean is 24.66%, the Arithmetic Mean (or Average) is 30.0%. The Internal Rate of Return is -4.6%.
In Example 2, it is important to note that at the end of each set (A,B,C) 100% of the initial investment is returned. So set A is simple IRR = 10%, if you do not return the initial investment you have a negative IRR since you invested 100 and only got back 10. If you invested 100 and received 10 in T1 and we assume the initial 100 is safely out there somewhere to collect, then that is Yield not Return. Return can only be calculated at the conclusion of the investment.
The IRR of set B is 27.6% and IRR of set C is 28.2%.
If you just want to see what your average yield is, which sounds more like the question is asking then we have to clear up another detail in the example. The example says there is a factor of 1.10 for T1 and 1.60 for T2. This suggests that the cash from T1 (10) is being added to cash fro T2 (50) which is the only way you get to 1.60%. The IRR for this scenario would be -24.11%, the yield in T2 is 60%.
Now in the example year 3 is awkward, it says the factor is 1.20. The problem with this is it doesn't really fit the other two factors (1.10, 1.60). Year 3 earned 30 and the factor is 1.20. Well if year two is 1.60 which we can see looks like 10+50 then how do we get 10+50+30 = 1.20? Shouldn't the factor be 1.90? If the factor at T3 is 1.20 it would mean we lost money since at the end of T2 we were up 60 and at the end of year 3 we only have 20, so we lost 40.
If we just wanted to know what the overall yield is for the investment we would sum the cash flow (10+50+30) and divide by the investment (-100) which is 30%. In the event you had a greater gap between each years cash flow say like (10+150+30) then Geometric Mean of 35.57 is a better indication of the end result opposed to the Arithmetic Mean which is 63.33. It is the same as weighting the averages of going to the store and buying bread, milk and caviar. The caviar as an out layer to the population will push the simple average higher.
Good thread and discussion!
What would you use if you have net worth gains weekly for the last 7 years?
"Net worth gains weekly"? While I understand what you are driving at it seems this likely not a benchmark. Does your inventory move that much from week to week?
I suppose a monthly review might ("might") be relative but it would depend on your volume. If the reason for such volatility in the number on your balance sheet is say your bank balance which is going up and down based on your cash in and out puts then you are measuring the wrong thing and the data is not all that important but I suppose it might be interesting to know.
Again if you use the rule of you only have a gain when the investment concludes then you start to get to a homogeneous set of data to compare to.
Why are you not focusing more on transaction returns which you compare returns or cash flow yields over the seven years?
To me the example means that those are the numbers at the end of each year. Cash flows have no place in the discussion because they are not given as inputs. Thus an IRR doesn't seem to be a choice of measurement unless I am missing something.
My assumption is that the worth of the investment at the end of each period is being compared with the worth at the beginning of the investment to arrive at a percentage. The goal is then to take those percentages and figure out how they grew on an average basis. Is anyone interpreting this differently?
My assumption is that the worth of the investment at the end of each period is being compared with the worth at the beginning of the investment to arrive at a percentage. The goal is then to take those percentages and figure out how they grew on an average basis. Is anyone interpreting this differently?
I didn't really follow you there. "To me the example means that those are the numbers at the end of each year." So, that is the NET cash flow at the end of each year, right?
The "worth" of an investment? How are you figuring this out?
"Worth" suggests you still own it, at the least, I would say this is sort of pointless to do more than once or twice a year at best. Seems more like an ego play opposed to an operational measurement.
You can't realize a gain until said gain occurs so what is the point of net worth outside of just talking point?
If "Worth" is being compared to a measurement like AUM (Assets Under Management) then you are going to use either your total purchase monies or total collateral values. The volatility of this number will depend on how active you are buying and selling new properties. Even still, it does not speak to how well you are managing anything. My AUM (Net Worth) can increase by 50% which really only means I bought a bunch of stuff not that I managed it well.
I am looking at my overall worth at the end of the year and comparing them with the worth at the beginning of the year. This generates a percentage. I then do the same exercise each year. The question is which mean is more accurate. Based on what I have seen the geometric mean is more accurate. This is no different than what any financial institution does and it does not represent some ego-based goal.
I'm trying to keep this simple. Introducing cash flows would require untold amounts of work. Of course nothing is realized until it is monetized. However, one could argue that the cash is still an investment as part of an overall portfolio too. One could split hairs over what the dollars are actually worth because the purchasing power vacillates daily.
The goal is to get something relatively simple that is as accurate as possible. I fail to understand why using this figure would not be a pretty good indicator of how well money is being managed. Perhaps I am missing something. I am assuming you are talking about distortion in accounting, but if that is the claim then nothing could ever be measured. The only way anyone could ever measure anything would be to fully liquidate everything to "realize the gain" which clearly would be stupid.
Good discussion here...
If I had to guess, I would guess that this is the disconnect right here. I can't imagine any time-over-time balance sheet metrics that would be very meaningful me. Sure, it's fun to see how much more I'm worth today than yesterday, but in terms of operational metrics, it's just not meaningful.
Perhaps that is important to you, Bryan, in which case I would agree that geometric mean of the increases should give you the information you are trying to glean in its most accurate form.
Well, in assets such as real estate, which do not have mark to market capacity, the way it is accounted for is only dealing with the actual. As value is not true until liquidation is achieved.
Measuring your net worth only shows that you increased how many assets you own, it is not a bench mark of management. Consider the following:
T1 = 10
T2 = 50
T3 = 30
So in year two, what is illustrated that you acquired more assets, 40 more to be exact. So you had a 400% increase in your worth. So what? Perhaps someone gave you $1 Million to invest with. In year three you sold assets, 20 worth. So your net worth dropped by -40%. Again so what, you liquidated some assets. It is not reflective of what gains or loss you had, merely the fluctuations in your asset schedule.
You could have gone from 10 in year one to 30 in year three with millions in loss. Adversely you could have gone with millions in gains.
Financial institutions do the net worth calculation you are talking about in relation to AUM. Which is not relative to return in any manner. You can have 200 Billion AUM and have a return of -50%. In year one, say a fund opens and perhaps it only raises 500k and then in year two say it raises 10 million. It is important to understand the raise since it might reflect in the return if some of the money was not fully invested to generate a return. However, it is a footnote to the return number and is not a major talking point in the discussion. Return is return and yield is yield.
Including cash flow is what is suppose to be done. To not include the cash flow your just doing math on numbers. Return includes cash flow by definition. Yield includes cash flow by definition. Its not complicated, it is proper.
You can do what ever you want, it doesn't mean it is right. If you mentioned to me your NW was X, I would still ask you what returns you had. If you use the difference between year beginning NW and year end NW you simply would be wrong. It has nothing to do with distortion in accounting, what you are doing is just trying to create a meaning from something that doesn't mean what you intend it to mean.
Simplest way to say it....
The difference between NW at the beginning and end of a year is not return it is just the increase or decrease in your portfolio size. It is not the return you generated on the portfolio.
Perhaps you are getting caught up with some compounding of the investment. Whereas you say I took 10 and made it 50, great job! The 10 made a return, 400%, which allowed you to reinvest to 50.
The return is still what matters most.
I'm not following. Perhaps it is because I am dense. If I increase my AUM it does not mean I am increasing my net worth. That is where I am getting lost with what you are saying. I can buy $10M worth of new investments that are worth $10M and my net worth goes up $0.
Am I missing something?
Well some businesses don't buy and sell several times a year. The accounting for net worth is only meaningful because the asset has appreciated (or depreciated) or there is a net change in cash or outstanding debt. Waiting to liquidate would mean that one is not able to measure anything for 30+ years.
The relation I am trying to draw is that is not really a performance benchmark that is going to deliver back clear results.
Just as you say you can buy $10M of new assets, technically your AUM goes up $10m. Provided there is not a counterbalance liability the Net Worth would increase by $10 M.
Adversely, fluctuations in net worth can be derived from playing with the assets and liabilities a part from actually delivering a return.
But there always is a counterbalance. The accounting identity is OE = Assets - Liabilities. How would one create net worth (equity) by increasing AUM?
I can assure you this is not what I am doing. A decrease in outstanding long-term debt is a legitimate increase in equity. The accounting profession may not like marking real estate to market, but that does not mean that one should not do it because the accountants don't like it. The alternative is to carry something on your balance sheet at what is clearly an incorrect valuation for years or decades, which is clearly more of a distorting factor than any error introduced by valuing it incorrectly at year's end.
We are starting to dilute the thread with this tangent. Net Worth is net worth and AUM is AUM. We can agree they are different things as they are.
In any discussion I am more interested in an actual return opposed to increase in net worth.
Yes...I do agree the thread is getting diluted, but I don't see how your reasoning tracks. Assets and liabilities are reported if there is an increase in AUM so that the change in equity is zero. Maybe I am missing something here.
I am interested in maximizing the growth of my equity. It still seems pretty easy to me to track this monthly or annually and use a geometric mean to figure out what the actual return is. I still fail to see why tracking an increase in net worth is misguided when trying to calculate a return.
I'm asking...hopefully someone can explain it to me.
Dion...I spoke to the folks that do the accounting today and each capital account from our fund would portray the individual investor's equity from the venture. So the problem you describe above would not be an issue with our fund due to the way it is structured. The liquidation preferences would not drive an increase in an individual investor's equity simply by raising more money.