Investments involve predictions about the future, and those predictions can be uncertain. The formal representation of that uncertainty is accomplished through the use of probability theory, such that every uncertain number X is now represented as the probability distribution function P(X).

Given two investments which give uncertain returns X and Y respectively, how does one choose between them? Under decision theory, a utility function U is used to compare <U(X)> and <U(Y)>, and the investment with the higher utility is chosen. For the sake of simplicity, modern portfolio theory models U using a quadratic function, where U(X) = X - r*X^2, with a greater r meaning more risk adversity.

Given a quadratic utility function, it is possible to evaluate the expected utility of all linear combinations of X and Y by just knowing <X>, <Y>, <XY>, <X^2>, <Y^2>. This result generalizes to any number of investments. Using this information, for any given return, one can find the linear combination which minimizes the variance, forming the optimal portfolio for that return. The set of all optimal portfolios forms the efficient frontier in the diagram below.

Now examine the effect of adding a risk-free asset. Taking a linear combination of the risk-free asset and any given portfolio, you can achieve any new portfolio with the same Sharpe ratio (ratio of the difference between return and risk-free rate to the standard deviation) as the given portfolio. The capital market line represents the best portfolios that can be formed this way - they consist of combinations of the risk-free asset and the market portfolio, which is the efficient frontier portfolio with the largest Sharpe ratio.

Under the CAPM pricing model, it is assumed that stocks function as components for building market portfolios, and stockholders are only compensated for risk which cannot be diversified away by the other components. Approximating the market portfolio with the SP500, we approximate this non-diversify-able risk using the covariance between the stock and the SP500. That covariance is scaled by the variance of the market portfolio and called beta.

---

This is the standard classroom treatment of the concept of risk, and I am uncomfortable with it in ways that I will describe in the next post. For additional reading, you have wikipedia:

http://en.wikipedia.org/wiki/Modern_portfolio_theory

Given the choice between money today and money tomorrow, one would prefer the money today. This is true because money today can be hoarded until tomorrow, or alternately used for other purposes before tomorrow - "money tomorrow" is just one out of many options available to someone with "money today" and hence "money today" must be of greater than or equal value to "money tomorrow". Similarly, a greater sum of money is always preferable to a lesser sum, because I have the option of discarding a part of the greater sum to get the smaller sum. It is then plausible that there be a greater sum of money tomorrow (s2,t2) which is of equal value to a smaller sum today (s1,t1).

What is the relationship between (s2,t2) and (s1,t1)? So far we have found that (s2-s1)(t2-t1) >= 0. Only by assuming that investment opportunities are of much shorter duration and of much smaller size than the quantities under consideration, and also available equally throughout time (i.e.  having multiple instantiations at (t1+dt,t2+dt) for all dt), do we get the conventional discounting rule, where (s2/s1) = r^(t2-t1) for some r > 0.

To go from the inequality to the equality requires work. This specific equality only emerges because of the additional assumptions made. Given access to a pool (defined by our assumptions of time-invariance and scale-invariance) of investments with discount rate r, we can now arbitrage any cash flow to its present value. The net present value (NPV) is this concept, and when positive it represents situations in which choosing the investment over choosing the pool results in a cash gain.

So far, I have only dealt with the opportunity cost of money, as represented by participation in the pool of investments. I will discuss the risk of the investment itself next.

http://en.wikipedia.org/wiki/Two_envelope_problem

You are given two envelopes to choose from, and are told that one has twice the amount of money as in the other. You pick one and open it, finding $20. The host asks you if you would like to switch. It is apparent from the random nature of the initial choice that switching would not change the payoff.

The only possible amounts in the other envelope are $10 and $40. I became tempted to assign 2/3 probability to $10 and 1/3 probability to $40, so that the expected value came out to $20. This is wrong, however, because the host could announce that he would pay a square of whatever amount of money I ended up with ($100 for $10, $400 for $20, $1600 for $40) and still switching would not matter, yet those probabilities (2/3, 1/3) would prescribe switching.

This is apparently a state of ignorance in which the possibilities are finite and known, and yet probabilities cannot be assigned to them. This disturbs me.

HT: Lee Hsien-Yang's New York talk last Friday