In the previous chapter, we saw that finding a Markowitz efficient frontier in an equality-constrained setting was simple: just use Lagrange multipliers and specify a closed-form solution. But finding an efficient frontier got more complicated in Section 4.1.3 when the constraints were inequalities. The example we used there with only three assets was simple enough to think through explicitly.

In the previous chapter, we saw that markets in effect have moods – at times nervous, at times overconfident – that persist for a while but eventually revert to some long-term middle-of-the-road mood. We don’t find it unusual when our high-strung friend has a meltdown because his socks don’t match, but the same behavior in someone who is normally unflappable makes us sit up and take notice. It’s the sudden change from a calm mood to panic, and the eventual relaxation back to calm, that produces fat-tailed distributions.

Risk is lack of information about the future. A situation is risky if it has widely varying possible outcomes and there’s no way to determine with high confidence which outcome will occur. A riskless or risk-free situation is one whose future is known exactly.

Benjamin Graham famously anthropomorphized the US stock market, attributing wild emotional swings to “Mr. Market.” Sometimes Mr. Market was fearful, usually after a sufficiently traumatic negative event.

In the previous chapter, we saw a variety of distributions that can be used for situations that fall into Knight’s a priori risk category. If we are confident that a known distribution describes all the outcomes and all the associated probabilities for a set of variables, then we might even be able to get a closed-form description of relevant risk metrics for these variables.

Chapter 3 investigated the seeming impossibility of uncertainty arising from certainty: even when we know exactly when and how much money we will get in the future, the value we assign today to that future money can be highly variable.

A hedge is a strategy intended to remove or lower a source of uncertainty about the future rate of return of a portfolio or company. The ultimate hedge is the creation of a risk-free arbitrage like the ones discussed in Section 2.7, although as we noted there it’s unlikely that profitable riskless arbitrages can last long.

A fundamental premise of security markets is that securities do not move independently of each other. If they did, then buying $n$ equally weighted securities would create a portfolio with variance equal to $\frac{1}{n}$ times the average variance, so any sufficiently large portfolio would effectively be risk free.

Fixed-income (UK Fixed-interest) investing is based on the translation of future money to present money via borrowing arrangements. A lender has a long (positive) position in future money (including any interest and principal payments) and a short (negative) position in present money. The interest rate that the lender receives reflects among other things the tradeoff of future money for present money.