Lessons in FM: Part III - Risk

Note: This is a continuation of the series Lessons in Financial Mathematics. Please read Part I - Present Value and Part II - Annuities if you have not already done so. They will aid greatly in your understanding of subsequent parts of the series. Although for this article, you can probably get along just fine without reading those if you really don't want to.

Risk. You've probably heard the term a lot in various contexts. Around the SL capital markets, you've likely heard of it as far as business risks, risks of default, and/or risks of de-listing. I know of one company which specializes in insurance in SL, The Rock Insurance, which is a tool used to mitigate risk. But has anyone ever defined risk for you? Beware, however, because this is a mathematical definition of risk. Finance professionals probably wouldn't accept my definition, or only grudgingly so, because of the consequences it brings. Here we go, though:

Risk = Standard Deviation(Returns)

That's it. Anticlimactic, I know. Mathematically speaking, we measure risk through standard deviation (or variance, the square of standard deviation), which can usually be estimated. Please note that this means that risk occurs regardless of whether the security in question goes up or down. Obviously most investors would rather have the risk only go one direction, up, but such is life.

In order to accurately gauge risk, we need a measure of earnings without any risk. In my classes, we defined a risk-free asset, usually United States Treasury Bills (T-bills), to be the risk-free assets earning the risk-free rate of return. Take a look at what they're currently earning here. Note that these are annualized yields - you don't really earn 2.98% in 4 weeks' time.

Anything above that risk-free rate is defined to be a risky asset. How much investors need to be compensated in addition to the risk-free rate to invest in that asset is called the risk premium, and can be computed as follows:

Risk Premium = E(Return of Asset) - (risk-free rate)

Note that I've used the notation for expected value (E(X)) here, because the return on the asset in question is risky - it will vary. By definition, the asset determining the risk-free rate of return has no risk.

So let's take a look at the risk premium for a typical bank deposit in Second Life. The SLCapEx is currently offering rates of 0.1% compounded daily. To make that an annualized rate, we take

[(1 + .001)³⁶⁵ - 1] = .44025

or 44.025% return. But, we're not done here. The SLCapEx cash accounts are a risky asset, and there is some probability that they will not pay out. We've already seen several bank collapses in SL this year, and that's the probability I'm talking about. I'm going to assume, for illustrative purposes, that the SLCapEx has a 90% chance of paying that rate of return, and a 10% chance of paying zero. Therefore, we calculate the expected value of SLCapEx returns as:

(.44025)*(.90) + 0*(.10) = .39623 = 39.623%

The risk-free T-bill had a rate of return of 2.98%. Therefore, the risk premium on a SLCapEx cash account is 36.643%! That is pretty staggeringly high, and usually those risk premiums are found only on the Las Vegas Strip.

So why aren't hordes of billionaires coming into Second Life to take advantage of SLCapEx's great rate of return? They're afraid of the risk. To them, the risk of losing their capital (and they would use more sophisticated models than I assumed above) outweighs the potential return that they could get. They are (as are most investors) risk-averse, meaning that they prefer less risk to more risk. There are some who are risk-takers, that will choose a riskier asset to a less risky asset. The third option is being risk-neutral, which means you don't care about the risk and just look at the rate of return, and I believe it's the rarest case of the three.

At this point, some pictures might be helpful. What we need is a graph to help us understand the relationship between risk and return. Fortunately, one exists, and is called the efficient frontier. What you do is you graph risk (measured either by variance or standard deviation) on the x-axis and rates of return on the y-axis. Then, by economic argument (not mathematical proof, mind you) you only take the highest rates of return for given levels of risk. What you wind up with looks something like this:


Pretty, isn't it? That line represents the best that you can do in the market. The lower dots are ignored, because investors would choose the higher dots (securities with a greater rate of return for the same risk) instead. Think about it - if you had two banks, one with a savings account at 3% and the other at 5%, you'd probably go to the one with the 5% rate of return (ceteris paribus).

There are an infinite number of points on that curve, however. How do we decide which one to take? Well, that's where your individual preferences come in via something called utility curves (or indifference curves). Actually, these are multi-dimensional functions that are often projected onto two-dimensional graphs as curves. Anyway, these curves quantify how much return you must be compensated with in order to take on an additional unit of risk. You wind up with a picture that looks like this:



You should choose the portfolio that maximizes your utility and lies on the efficient frontier. Simple, right? Well, the mathematics gets messy, but ideally (economists would like to think) we do this every day without even thinking about it.

All of this stuff falls under the umbrella of modern portfolio theory, and I encourage those interested to do some exploring on Google to see the range of topics out there.

Next week - equivalent portfolios and their magic.

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Just a quick note: I take requests here at Second Chaos. If there's some mathematical, financial, or economic topic you'd like me to explore, let me know. I might refuse topics that I feel are too far advanced for me, but I'll gladly make articles explaining standard deviation or variance or anything like that. Just FYI.