Risk, wealth, utility — in one picture
The question the paycheck left open
The hundred-year price sample paid about 8.43% on average (it’s here if you want to read it). The miss around that average was about 18.58%. That’s a noisy payout. That’s the cost of doing business with Mr. Market. A household that treats a positive +18.58% and a miss of -18.58% as a wash is doing simple arithmetic and stopping there. But we know that the mathematics of gains and losses aren’t symmetric. A 50% loss requires a 100% gain to get back to breakeven We wrote about that here.
Dollars are also not the whole score. Dollars measures wealth and the size of your book. Wealth itself has its own score; it is called utility.
In the first primer (found here) we explored three concepts: risk, wealth, and utility. Utility, in this context, means the usefulness of wealth and its ability to buy things. Most important of which is the freedom to choose.
In this post, we put all of that in charts (and a little bit of math). Let’s do it.
Wealth
Wealth is the stock. We measure it in dollars. Get enough of it and you get a book. The objective is to grow the book over time.
We do not accumulate dollars because the number is pretty. Well, it is but I digress. We accumulate dollars because they buy a life that is not hostage to next month’s invoice. For a lot of us that’s a mortgage, a car payment, a grocery bill.
It also buys a retirement plan, the right to walk away. Call it freedom. That is usefulness. That is why the stock matters. That is why wealth matters.
As John Goodman famously quipped in the movie The Gambler, “that’s your fortress of fucking solitude”.
That’s your “f*** you money”. That’s your base.
Utility
You build your base because its useful. Wealth is useful. In economics that usefulness has a fancy term, they call it Utility. One thing to keep in mind, the “Usefulness” or ‘Utility” of wealth isn’t a straight line. The first set of dollars are there to keep the lights on and pay the basic bills. Later dollars still help and are still good. But they help less. A household that already has a hundred thousand does not feel a twenty-thousand bump the way a household climbing off zero does. Same dollar. Different usefulness. One household has enough just to cover basic needs, the other has some wealth left over.
So, when accumulating wealth, always make sure you have more than enough to cover basic needs. That’s the first two bottom tranches in Maslow’s hierarchy of needs (physiological and safety needs). And when you do it, do it within your means. Not everyone needs a mansion to build a comfortable home. The residual income is what builds the base. And when you have a solid base, you get to work your way up the remaining three tranches (belongingness, prestige, self-actualization).
This is why wealth is important. You need it for its utility. It’s a tool in the same way a hammer is when nailing things down.
And since it’s a tool, we need a way to measure its performance. That performance is a score and we write it as u(W). Where W is wealth as measured in dollars; u is a function of W and it measures how much that wealth does for you.
The way to keep track of the score as you make it is simple. It’s the natural log.
u(W) = LN(W)
The line rises. It rises more slowly as W (x-axis) gets large. The slope at any point is how much extra usefulness the next dollar buys, that’s the first derivative u’(W). That slope is also falling, the change in that slope is the second derivative u”(W).
That is the shape of the score. That is diminishing marginal utility. It isn’t symmetric. It is the same mathematics that describes why the pain of a capital loss is a greater emotion than the joy of a capital gain. What you thought was a mood is actually just math.
Diminishing marginal utility. Sounds like another nerdy term concocted by eggheads with nothing better to do. Stay with me as we pull on this thread a little bit longer.
Let’s say you walk into a random guy on the street. Let’s further assume that the random guy is someone’s drunk insolvent Uncle Sam who has no idea how to live within his means and can barely keep a couple of dollars in his pocket (topic for another day).
He proceeds to tell you a sob story on how he only has $10 dollars under his name. You instantly take pity and you give him a buck. Let’s call this generous $1 contribution ΔW. His new wealth (or lack thereof) then increases by:
He is now 10% richer just by meeting you. Whether or not he earned it is beside the point.
Now compare and contrast Uncle Sam’s sob story to yours. Let’s say that you’re a person of “not-so-limited means” because of discipline and hard work. Let’s further assume that you have $100k in your bank account. Let’s walk through the math.
What this demonstrates is the fact that the utility of an additional dollar to you is meaningfully and significantly less than that that of Uncle Sam’s. A thousandth of a percent, in fact.
Same curve, two different scores, two different slopes, u’(W). And that slope decreases the further to the right you go on the curve u”(W).
This in a nutshell is utility; and utility is how you score wealth.
Risk aversion
Two down, one to go. Risk aversion. Stay with me.
Let’s propose another helpful analogy. No Uncle Sam on this one. Look at risk aversion as if it’s a game with a short menu. The menu has two offers with the same expected dollars:
One item shows a sure reward of $100,000.
A second item is a flip of a fair coin (50/50). A tail shows up and you get $80,000 while a head turning up gets you $120,000.
The expected value of the second menu item is:
EV = ½ EV1 + ½ EV2 = ½(80,000) + ½(120,000) = $100,000
Again, two different choices, same expected values. One risk-free, the other not so much. You being an astute but more “risk-averse” investor will take the sure thing. But if you decide to take the second choice for the chance of a higher payout, then you are not as risk-averse as you claim to be because this higher payout has a downside pair and you’re willing to take it. It’s not a sure thing
Remember that equation for utility LN (W)? That’s not a straight line. If usefulness were a straight line, you would take either choice. You would shrug it off because there’s no difference in utility.
But you can’t because you know there’s a difference. You take the first choice, the sure $100,000. That preference is your risk aversion. It is not “I hate losing money”. Everyone hates losing money. It is not volatility. At least not yet. Risk aversion right now is just a preference, a preference to take a sure thing over a bet.
The accounting book shows equivalent values, they’re both $100k. But in wealth and utility, they are two different things. One sits on the utility curve because it’s a sure thing, the other does not because the outcome hasn’t happened yet. You have to flip a coin (Bernoulli trial). And that trial, in real life, has penalties.
We’ll show you the math behind it.
The picture
First, you draw a curve LN(W), that curve is the usefulness of sure wealth. You calculate yours by scoring the sure check on the curve: LN (100,000). Real wealth sits on the LN (W) curve. We know it’s utility because it’s real. You don’t’ need to flip a coin.
The flip of the fair coin also has utility. You get it by averaging the two outcomes: half of LN(80,000) plus half of LN(120,000) because it’s a trial. It’s called Bernoulli. You have to flip a coin if you chose the second option. It also has an expected value and you need it for the x-axis. The expected value is also the average of the low (80k) and the high (120k) payouts. That midpoint is $100k.
The sure check will score higher. You pick it because you’re risk averse. And since its real, it sits on your utility curve.
You determine the shape of the curve by computing the following numbers:
U1(W) = LN(80,000) = 11.2898
U2(W) = LN(100,000) = 11.5129
U3(W) = LN(120,000) = 11.6952
Take the points above and line it up against $80,000, $100,000, and $120,000 on the x-axis. The sure check is the point at $100,000. That’s the choice you made the sure check. It’s utility u(W) is 11.5129. That’s the score.
You draw a second line (dashed line). You connect the dot from $80,000 on the x-axis to the other dot on $120,000 and you find the mid-point. The mid-point should be $100,000 on the x-axis. Find the corresponding value on the y-axis and you get the corresponding u(W) for the dashed line. It’s not on the solid curve because it’s not real wealth yet so it sits on the chord (dashed line). The resulting expected utility should be slightly below that of the sure bet.
Sure $100,000 sits on the curve. The 50/50 coin of $80k/$120k sits on the chord. CE ≈ $97,980.
At $100,000 the curve sits above the chord. That vertical gap along u(W) is usefulness, not dollars. The coin and the sure check have the same expected dollars as seen on the x-axis but they do not have the same score.
Slide left along the wealth axis until the curve has fallen to the height of the chord at the midpoint. That wealth is the certainty equivalent — CE shown as a green dot. It is the sure check that feels as good as being forced to flip a coin. On this coin, the certainty equivalent is about $97,980. Worth slightly less than the sure thing because it should be.
The horizontal gap is the cost in dollars. $100,000 – $97,980 = $2,020. The coin flip is only worth $97,980 as a sure check. That $2,020 is the risk premium — or, said another way, what you would pay to swap the coin for $100,000 cash. You do not pay $2,020 to play. You mark the coin flip down.
Not symmetric
From $100,000, a move down to $80,000 is a $20,000 hole. A move up to $120,000 is a $20,000 bump. As you can see below, the dollars may be equal but the usefulness is not.
LN(100,000) − LN(80,000) is LN(1.25) ≈ 0.223.
LN(120,000) − LN(100,000) is LN(1.20) ≈ 0.182.
The move down takes more usefulness than what the move up returns (remember the pain of a loss versus the joy of a gain?). Average the two moves and you sit below the sure check. That is why the chord is under the curve. That is why CE is left of $100,000.
A dollar lost is not the flip side of a dollar gained — not because the dollars differ but because the usefulness of those dollars differs.
A number named A
How strongly the score flattens can be written as one number:
A = −u″(W) / u′(W)
At A = 1 the score is LN(W). That is the picture above. Raise A and the same wealth axis hunches more; the premium on the same coin gets larger. Lower A and the line straighten[vd1] s out; the shrug comes back. We will measure A later.
What this isn’t (yet)
This isn’t your personal score. That will come later. This is an example on what the concepts are. We’ll keep coming back to it.
As you go through this journey you will learn something about yourself and you’ll start asking the right questions. What’s my degree of risk aversion? You’ll get to the right answer. And no, it’s not “I hate losing money”.
Stay tuned!
