efficient frontier and portfolio selection
When you mix investments together, you choose not just how much you expect to earn but how much you might bounce around — the risk. Plot every possible portfolio on a chart with risk along the bottom and expected return up the side, and they fill a region. The efficient frontier is the upper-left edge of that region: the set of portfolios that give the highest expected return for each level of risk (or, equivalently, the lowest risk for each level of return). Any portfolio below the frontier is wasteful — you could get more return for the same risk, or the same return for less.
The key insight, due to Harry Markowitz, is that risk is not just about each holding on its own but about how they move together. Combining assets that do not rise and fall in lockstep reduces the wobble of the whole portfolio more than you might guess — this is diversification, the one 'free lunch' in investing. A simple picture: two assets each fairly risky on their own, but which tend to zig when the other zags, can be blended into a portfolio that is calmer than either, while keeping a decent return. Working out the frontier means estimating each asset's expected return, its volatility, and how each pair moves together (their correlations), then finding the best blend for every risk level.
Why it matters to actuaries: portfolio selection is the high-level question of how much to put in bonds, equities, property and so on, and the frontier is the textbook frame for it. But an insurer's version has a twist the original theory ignores: the goal is not to sit on the frontier in isolation but to choose the portfolio that best backs the liabilities, so the relevant 'risk' is the risk of failing to meet promises, not just the volatility of asset value. The honest caveats are large: the frontier depends entirely on estimated returns and correlations, which are noisy and unstable, and correlations have a nasty habit of jumping toward one in a crisis — exactly when diversification is supposed to help, it can vanish. The frontier is a way of thinking, not a precise map.
A fund considering a 100 percent bond portfolio finds that shifting 20 percent into equities lifts expected return with only a small rise in overall volatility, because stocks and bonds do not move in perfect step — a move onto the efficient frontier.
Diversification can lift return per unit of risk — until correlations break in a crisis.
The frontier is built from estimated returns and correlations that are noisy and unstable; in a crash correlations tend to spike toward one, so diversification can fail right when it is needed most.