## Sunday, October 21, 2012

### Rate of Return and Risk: The Analogy to a Power Plant & Battery System

In this post, my plan is to lay out an analogy for understanding rate of return on investment and risk, using batteries and power plants as an analogy. The goal is to answer the following questions:

What is the definition of a power plant?
What does it mean for a power plant to have a positive rate of return on work invested?
How do we quantify risk?
What are the units of measure for risk?

Rate of return on investment of a power plant and the battery analogy
Let's begin to answer these questions by imagining that we start out with a battery that is not fully charged, but not completely drained either. Let's say that the maximum storage energy is 1 MWh and we currently have only 200 kWh of energy stored in the battery. This means that we have 200 kWh of useful work that we can spend. (Let's imagine also that there are no rules against fully draining the battery.)
So, what should we do with that 200 kWh of useful work? Should we spend it on building houses, cars, or TV's? If we spend it all on these items (as well as a whole list of items I can think of), then we will have spent all of the 200 kWh and we would have completely drained the battery. We will have no "useful work" left in the battery.
Instead, we could spend half of the 200 kWh of stored electricity by giving 100 kWh of electricity to companies that drill natural gas wells and that build natural gas combined cycle power plants. We can either give the electricity to the company in a form of a bond (in which the company agrees to give us a certain return on investment after X years) or in the form of stock (in which we become co-owners of the plant and get paid in dividends if/when there are profits.) In the case of bonds, after X years, the company may agree to pay us 200 kWh of electricity. In the case of stocks, as long as we chose a good company, the company will pay us (in electricity in this analogy) and the company will likely pay us more than 200 kWh of electricity in dividends after X years. So, if we spend our electricity on building natural gas wells and natural gas power plants, then after X years (where X is likely between 4 and 12 years in this analogy), we will have increased the amount of stored electricity in our battery. If we invested 100 of the 200 kWh, then after X years we will likely have more than 300 kWh of electricity. We could spent 50 kWh of electricity on goods, such as clothing, food, entertainment, etc..., and we would still have more than 250 kWh of electricity in the battery. [Note that there is no return on useful work when you invest in clothing, housing, and entertainment. We clearly need clothing, housing and entertainment, but these activities do not grow our economy, i.e. our capability to do useful work. These activities are necessary because humans run power plants and humans need clothing, housing, and entertainment.]

When we invest money in power plant companies, the company is basically agreeing to give us some of the electricity produced at the power plant in exchange for our investment into the initial capital cost of building the power plant. This cycle of investment of electricity and production of electricity can continue as long as there are exergy sources that yield positive rates of return on work invested (which will be the case for at least the next billion or trillion years.) This is the cycle of growth. You invest useful work into projects that increase the total amount of useful work. We started with 200 kWh and now we have more than 300 kWh of useful work, from which we can spend between 0 and 100 kWh of electricity during that time period on 'goods' and still have more electricity than when we began.

Definition#1: The technical definition of a power plant is a project that has a positive rate of return on useful work invested

Note that the definition of a power plant is not a device that creates electricity. A power plant is a project (i.e. a cycle) in which the investment of electricity into the cycle creates more electricity (after one full loop through the cycle) than was invested into the project initially. Sadly, many of the power plant designs I see people discussing right now are not power plants by the definition listed above. Most new solar thermal, solar PV, wind, and coal with >90% CO2 capture projects are not actually power plants because they consume more useful work building them than they generate over the ~30 year lifetime of the project. On the other hand, investment into new natural gas wells in the US and new natural gas power plants can yield large positive rates of return on useful work invested (>20%/yr). While right now most new solar thermal, solar PV, wind, and coal with >90% CO2 capture projects are not actually power plants, this doesn't mean that they can't in the future become power plants. [Note that these projects don't become power plants when the price of electricity increases; they could become power plants if the capital costs (as measured in used work spent) decrease, such that the project consumes less useful work in upfront capital and consumes less useful work in reoccurring fuel and maintenance costs than the amount of electricity it generates over its lifetime. Note also that this technical definition is not the economic definition of a power plant. The economic definition of a power plant can be more or less strict because the economic definition of a power plant also accounts for energy arbitrage, i.e. the fact that electricity prices are not the same at different points in time and space. We will not go into energy arbitrage in this post.]

So, the definition of a power plant with a zero percent rate of return on useful work invested is the following: If you start with 'Y' kWh in a battery and invest the 'Y' kWh in capital costs of the project, then a power plant with zero rate of return on investment is one such that, if all of the net electricity from the power plant goes back into the battery and if the battery doesn't lose stored energy over time, then after the power plant is closed down, there will be exactly 'Y' kWh of electricity in the battery...i.e. you break even and there was no point in building the power plant. Net electricity mean here means the net electricity generated at the power plant minus the amount of useful work required to fuel and maintain the power plant, i.e. fuel and O&M costs as well as any costs to purchase pollution credits if you have to purchase credits or pay taxes to emit pollution.

The definition of a power plant with a positive rate of return on useful work invested is the following:  If you start with 'Y' kWh in a battery and invest the 'Y' kWh in capital costs of the project, then if all of the net electricity from the power plant goes back into the battery and if the battery doesn't lose energy over time, then after the power plant is closed down, there will be more than 'Y' kWh of electricity in the battery...i.e. you increase the amount of useful work that you can spend in the future.

Now, I'd like to step back and address the question: does this mean that we should only be investing in power plants? The obvious and correct answer is:   No

My point here is not that we should be investing only power plants. It's the following:
We (as individuals acting through our investment fund companies) should be investing in power plant systems with rates of return on work invested greater than 10%/yr if we were to reinvest all of the profit back into building new power plants. But we shouldn't invest all of the profit back into building new power plants. Part of the profits need to go to the government (to carry out required functions, such as maintaining the legal courts, maintaining the police/FBI/military, maintaining roads, and conducting international diplomacy.) Part of the profits need to go into raising families (food, clothing, education, shelter, entertainment, etc...) and part of the profits need to go to collective needs that individuals are unlikely to fund themselves (such as large-scale science projects, searching for life on other planets, space exploration, and space colonization.) And finally we should leave enough of the profits so that we can reinvest the profits back into power plant projects. We should be aiming for net growth rates of at least 5%/yr. This means that, in order to fund the government, family development, and science/space exploration, we need to invest in power plant projects of at least 10%/yr rates of return on investment, and probably more like 20%/yr.
You can't grow the economy by investing in projects that consume more electricity than they generate over their lifetime (such as solar PV as of 2012....this may change in the future.) Likewise, you can't hope to grow the economy by investing in projects (like nuclear power plant) that only have rates of return on work invested of close to 4.2%/yr. (You can estimate this value in the US by using the estimated capital cost of \$6000/kW, assuming a 90% capacity factor, assuming it takes 6 years to construct the power plant, assuming that the capital costs are equally split across those 6 years, assuming a 30 year lifetime after construction, ignoring fuel/labor/maintenance, and using the price of electricity used to estimate the capital costs of the nuclear power plant...which is the price of average generation price of electricity that is paid to baseload power plants like nuclear power plants...roughly \$50/MWh.) A value of 4.2%/yr means that there is no way to obtain net growth rates above 5%/yr if you only invest in nuclear power plants. (I would call a nuclear fission power plant a marginal power plant. Note that if you assume a 60 yr lifetime, the rate of return on investment only increases to 5.5%/yr. Changing the lifetime from 30 yrs to 60 yrs has very little effect on the rate of return on investment.) If we want to maintain our standard of living, to continue to invest in large-scale science projects, and to explore other planets, then we can't invest in power plants whose rate of return on work invested is only ~5%/yr.  We need to invest in projects with rates of return on work invested greater than 10%/yr, so that some of the profits can go into all of the things mentioned above.
[Note this is why governments can't stimulate the economy through stimulus that goes into wasteful projects. A wasteful project is a project in which we decrease the amount of electricity in the battery after one complete cycle of the project. Examples of wasteful energy-related projects are cash-for-clunkers and DOE grant-funding for ethanol as well as solar, nuclear, and wind power plants. In these, as well as many other projects, the government spent useful work (after taking it from the citizens in the form of direct and indirect taxes) on projects with negative rates of return on work invested. These projects did not generate more electricity than they consumed in upfront capital. And these projects did not help maintain roads, police, military, legal system or space exploration.]

So, now we can hopefully see that we need to be earning rates of return on investment greater than 10%/yr so that we can afford to pay taxes, raise families, and invest in exploration, while maintaining net growth rates of 5%/yr in our capability to do useful work.

Risk of an investment and the battery analogy
We will now shift gears and discuss risk.
Risk is normally defined in the investment community as the standard deviation of the yearly rate of return on investment. The standard deviation is a unitless quantity. In the investment community, a risky investment is one in which there is a large fluctuation in the yearly rate of return on investment, perhaps some years it is 10%/yr, some years it is 0%/yr, and some years it is -10%/yr. A riskless investment is one in which the rate of return is always the same, perhaps 3%/yr.
This is one definition of individual project risk. I'd now like to step back and look at what it really means for a portfolio of projects to be risky. When we normally use the word risk, we mean the following:  "There is the risk that my family might not be able sustain our current levels of consumption...i.e. housing, food, education, clothing and entertainment for my family."

How does this relate back to the battery analogy?
First, let's assume that the current consumption of useful work (i.e. electricity in this analogy) by your family is 5 kW. This means that in order to continue at this rate of consumption of useful work for one month, then your family needs to have roughly 3.6 MWh of electricity stored in a battery that doesn't lose any charge over time.

But let's say that you have actually have 100 MWh of electricity to your name. Perhaps, you have 90 MWh of electricity invested in companies (such as power plants), and you have 10 MWh stored in a battery (that doesn't lose charge over time.) I would define your risk as  5 kW / 10 MWh,  i.e. 0.0005 per hr, which is also the same as 36%/month.

Definition#2: Your portfolio risk is the power you and your family consume on average [kW] divided by the amount of stored useful work [kWh] that can be accessed within the time period before you run out of useful work (i.e. money). The units of portfolio risk are [per month].

In the case listed above, if all of your investments were to fail at the same time, then you could continue to support your family for 3 months (ignoring other sources of income.) Here, we can see the relationship between total portfolio rate of return and portfolio risk. If you try to increase your total portfolio rate of return by increasing the amount you invest from 90 MWh to 95 MWh to 99 MWh, then you increase your risk from 36%/month to 72%/month to 3600%/month (which means that the total time you could continue supporting your family at the current rate of consumption would decrease from 3 months to 1.5 months to 10 days, if there were a failure in all of your investment.) Worse, you could actually have so much risk that if the investments were to fail, you would owe people more electricity than you actually stored in your battery. This means that you couldn't sustain the current rate of consumption for even a single second.

Note that risk has the same units as rate of return on investment (inverse time.) I think that it's important to measure portfolio risk in the same units as rate of return because then we can see how we can increase the rate of return only by increasing the portfolio risk. (There are many different types of risk that one can measure; this is just one type of risk.) Increasing portfolio risk means decreasing the time that you can sustain your current level of consumption. (Note that unemployment insurance is basically a way of lowering risk, but that this can only be achieved by a decrease in overall rate of return on investment. We, as society, have determined that it's better to have lower risk, even if it means lower overall rates of return on investment. I'm not going to argue for or against unemployment insurance. I'm simply hoping to point out that this way of lowering risk can only come about by decreasing the amount of electricity we invest in companies because we have to increase the amount of electricity we keep stored.)
One of the major problems we face today is that we think that we are storing useful work, but we actually have very little stored useful work. The only stored useful work we have is the hydraulic energy behind a dam, the stored gasoline at gasoline stations, and the stored petroleum in the Strategic Petroleum Reserve. We have virtually no storage of electricity in our society, which means that we are constantly living on the edge of a blackout. We are always potentially a few minutes away from a black out. I don't mean to state this to scare people (especially if you live near a hydro-electric dam with days worth of electricity storage.) I'm just trying to point out the fact that there is a huge discrepancy between what we think (i.e. we think that we have lots of money in checking/saving accounts that are '100% safe') and what is actually happening in the real world (i.e. we have virtually no stored electricity...only stored hydraulic energy and chemical fuels, such as gasoline for cars and food for people.) The reason that we can't all withdraw our money from our checking and savings accounts is that banks aren't actually storing our useful work in batteries or petroleum reserves. Banks are keeping a small amount of reserves in the form of gold/silver/food/hydraulic energy/gasoline storage, but you can't use gold and silver to do useful work. This is the main reason why I think that the U.S. Federal Reserve should sell its gold/silver reserves and should use the sale of gold/silver to industry to purchase items like petroleum reserves that can be used to generate useful work and that maintain their capability to do useful work over lone time periods (years). Batteries don't actually have long-term storage of useful work because they drain slowly over time. Though, with that having been said, I do think that it's a good idea for electricity grid companies (that aren't close to hydraulic dams) to purchase batteries to increase the storage capacity of the grid because many of us live in many places with only minutes spinning-reserve and only seconds worth of battery reserves.

In other words, in order to decrease our risk of blackouts, we could purchases batteries. But this most likely means decreasing our rates of return on investment. Batteries don't create useful work; they reduce risk by increasing the amount of time that we can continue current consumption. ([Batteries can also be used for energy arbitrage, which means that there may be economic reasons to purchase batteries, but I don't want to get into that in this current post because I'm trying to focus this post on the following points: (1) power plants are systems that increase the amount of useful work, (2) batteries are devices for reducing risk in the case of power plant failures, and (3) you can increase the rate of return by choosing better projects or by increasing your risk, i.e. decreasing your stored electricity and investing a larger portion of your savings into projects with sunk capital costs.]

The percentage decrease in total rate of return on investment caused by purchasing batteries is relatively small compared to the percentage decrease in risk by having around an hour of electricity storage in batteries (compared with ~15 minutes of spinning-reserve.) For example, power plants in the US generate roughly 500 GW of electricity on average. To have 1 hour of storage would require 500 GWh of storage. Since storage costs are roughly \$400/kWh, then it would take roughly \$200 billion to purchase each hour of battery storage. Purchasing one hour of storage corresponds to a roughly 1.5% decrease in the total economy (\$14 trillion), so the question is: are we willing to decrease our societal rate of return on investment by 1.5% (i.e. from perhaps 1%/yr growth to -0.5%/yr growth) in order to decrease of risk of a blackout? (There is no right answer, but clearly storage is not cheap.)

We now can address the question: should I leverage my investment by borrowing money to increase my expected return on investment?
What you define to be safe investing depends on your appetite for risk, but risking more money than you actually have is almost never a smart move. It's normally a recipe for disaster. For example, hedge-fund companies like Long-Term Capital Management L.P. (LTCM) thought that they could make riskless, leveraged investments because they assumed that the assumptions were valid in the Black-Scholes equation for pricing options (i.e. stock prices have Gaussian fluctuations in growth rates about a positive average value using past data to predict future rates and standard deviation.) This is not how the real world works. You can't use past data to predict the future, and the past data itself is not even Gaussian. There are fat-tail distributions in the fluctuations, which are indicative of Lorentzian statistics, and hence, you can't define the
A riskless, leveraged investments is an oxymoron statement. If you borrow 100 kWh to invest in a company and if you don't have 100 kWh sitting in a battery to pay back the loan, then you are taking on a significant amount of risk, and you might end up owing for money than you have stored.

Now, I'd like to step back and look at how we can apply the definitions listed above in our own lives when we make investment decisions.
Certainly, everything we do is risky, but once you start raising a family, you should have at least a few month of cash sitting in reserves (without taking into account unemployment insurance and possible ways to reduce monthly expenses.) We live in a society in which it pays to be extremely specialized, but specialization implies high risk, i.e. the more specialized you are, then the longer it takes to re-train yourself for a new profession. One of the main reasons to have unemployment insurance is to decrease the risk of being specialized. For the society in general, specialization has enormous net benefits, but it could also mean enormous individual risk because when you specialize so much, it doesn't take much of a change in the market for your specialization to become useless for the market.
This is one of the many trade-offs we face every day (should I take the risk of being more specialized by going back to school?
Another major trade-off we face is the following:  should I purchase a home or rent?  When you purchase a home, you are leveraging your small investment, and hoping that the value of the house stays the same or increases. But why should the price of a house stay the same or decrease? The value of a car decreases with time. Why shouldn't the value of a house decrease with time (after adjusting for inflation)? A house gets old and worn down. A house is not an investment in the same way that a power plant can be an investment. A house (like a car) can't generate more useful work than went into building it. The only way a house can make money (in inflation-adjusted dollars) is if you purchase a house in an area in which more people are trying to move into than are trying to leave. By purchasing a house, you become extremely leveraged. My suggestion is that you think really hard about whether you will save money purchasing a house (rather than renting) after you account for fees to real estate agents, local taxes, inflation, and possible depreciation due to normal wear and tear. One question is: if there is a net savings compared with renting, is it worth the higher risk? A house ties you and your family to a particular city, which can be a bad thing if your job is highly specialized.

In summary:
The wise investor is one who invests in power plant projects that earn rates of return on investment greater than 10%/yr, and then using the profits to (a) pay taxes to government, (b) maintain a good standard of living for his or her family, and (c) reinvest so that you maintain net growth rates of roughly 5%/yr.
The rule is pretty simple to state, but it's really hard to follow.
Why?
(1) There are always the sirens calling you to consume, consume, and consume!
(2) It's hard to invest in projects with >10%/yr because this requires actually having good information that is not already in the public sphere. If there were a project with a >10%/yr rate of return on investment, then it's likely that investors would have already spotted it and would have already loaned the company the money it needed for the project. Indeed, future expected revenue from the project is already likely reflected in the price of the stock/bonds, so investing in the company does not guarantee you a  >10%/yr rate of return.
(3) When we do have good investment information, we are likely to leverage too much, and take onto much risk.
(4)  Safe, riskless investments rarely increase in value faster than inflation.
(5)  It's impossible to predict the future.

So, in some ways, the solution is simple: invest in high rate of return projects, use only part of the yearly profits to fund your lifestyle, and reinvest the rest. But in other ways, the solution is not simple: we constant want to spend more money in order to make it appear that we are higher in cultural status than our income really would dictate.

The goal ultimately is to grow life, but there's no one right way to do this. The reason why I want to raise the level of education in the area of financial investment is that I want to see you all do well financially, so that life can continue to grow on this planet, and so that we can continue to invest in exploration of other planets. We know that we have grown in the past (right now we are not growing in the West.) The question is: can we learn from what it took to grow in past, so that we can return to grow in the future? We can't repeat the past, but I think that we can use the lessons and stories of the past to help us return to growth.