Take the two-asset arithmetic and simply plot it. Every weighting of Steady Staples (10%, 18%) and Cyclical Motors (14%, 28%) at ρ = +0.3 is a point on a risk-return chart that puts σ across and E(R) up:
| Weight in Staples | E(R) | σ |
|---|---|---|
| 100% | 10.0% | 18.0% |
| 90% | 10.4% | 17.2% |
| 78.6% (min-var) | 10.9% | 16.9% |
| 60% | 11.6% | 17.7% |
| 40% | 12.4% | 20.2% |
| 20% | 13.2% | 23.7% |
| 0% | 14.0% | 28.0% |
What makes a portfolio efficient or dominated?
Sketch it by hand once. It takes two minutes and the intuition is permanent. Notice that the points don't lie on a straight line between the two assets. Your curve bows to the left, pulled toward lower risk by the covariance term, and its leftmost point is the minimum-variance portfolio. Now look at the curve's lower branch, the portfolios holding more Staples than the 78.6% min-var mix, down to 100% Staples. Every one of them is dominated: directly above it on the upper branch is a portfolio with the same σ and strictly more return. No rational investor holds a dominated mix. The upper edge, from the min-var point up to 100% Cyclical Motors, is the efficient frontier: the menu of undominated portfolios, each offering the most return available at its level of risk.
With many assets the picture generalises. Thousands of possible portfolios fill a cloud, and the frontier is the cloud's northwest edge. Pinning it down for real takes the matrix algebra we're deliberately skipping. All the concept you need is the shape and meaning, a bowed edge of undominated portfolios, and the two-asset case computes it faithfully. Your risk appetite chooses where on the edge you sit. The arithmetic chooses the edge.
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What is the tangency portfolio, and why is it the market portfolio?
Now add one special asset and watch the picture simplify. Introduce a risk-free asset yielding Rf. Its σ is zero and its correlation with everything is zero. Mix it with any risky portfolio P and the combinations plot as a straight line from Rf to P. Risk scales linearly with the weight on P, because there's no covariance term to bend it. Different choices of P give lines of different slope, and the slope is the deal you're getting: (E(RP) − Rf)/σP, extra return per unit of risk. This ratio has a name you'll use for life, the Sharpe ratio. A rational investor wants the steepest line, the one from Rf that just kisses the frontier, at the tangency portfolio.
This leads to two-fund separation, the hinge between Markowitz and CAPM. Once the risk-free asset exists, every investor's best holding is some mix of just two things: the risk-free asset and the same tangency portfolio. Cautious investors hold mostly Rf plus a little tangency. Aggressive ones hold the tangency portfolio levered. Nobody's optimal portfolio differs in its risky composition, only in its dose. And if everyone's risky money chases the same bundle, then in equilibrium that bundle must contain all risky assets, weighted by market value, or prices adjust until it does. So the tangency portfolio is the market portfolio. In practice we proxy it with a broad index. For India that's the NIFTY 50 or NIFTY 500, and for the US the S&P 500. CAPM's scorecard questions that compromise.
Capital allocation line vs capital market line
Those lines have names, and the names are worth learning because the profession uses them to mean two different things. Any line drawn from Rf through any risky portfolio you happen to hold is a capital allocation line: the menu of risk-return combinations available to you, given that particular risky holding and a bank account. The one special line drawn from Rf through the tangency portfolio, which in equilibrium is the market portfolio, is the capital market line. Every capital allocation line is a real menu, but only the capital market line is the best one. A portfolio with an expected return of 9.5% and a standard deviation of 22% offers a Sharpe ratio of (9.5 − 4.2)/22 = 0.241, so its capital allocation line reaches 7.45% at a 13.5% standard deviation. At a Sharpe ratio of (8.52 − 4.2)/16 = 0.270, the market's line reaches 7.85% at the same risk. Same risk, four-tenths of a point more return, and the only difference is which risky bundle you started from.
How does your risk appetite pick a point on the line?
So the arithmetic picks the line. What picks the point on it? Risk appetite, and the standard way to write risk appetite down is a utility function:
U = E(R) − 0.5 × A × σ² (returns and sigma as decimals)A is the risk-aversion coefficient, typically a number between 1 and 10, where higher means more averse. Read U as a certainty equivalent: the guaranteed return that would leave this investor just as happy as the risky portfolio. The σ² term is the penalty, and squaring is what makes the penalty bite disproportionately as risk rises.
Maximising that utility along the capital market line gives the weight to put in the market portfolio:
w* = [E(Rm) − Rf] / (A × σm²)Work it for two investors facing the same market, with Rf = 4.2%, E(Rm) = 8.52% and σm = 16%. A moderately averse investor with A = 2 puts 0.0432/(2 × 0.0256) = 84.4% in the market and 15.6% in bills, for an expected return of 7.85%, a standard deviation of 13.5% and a certainty equivalent of 6.02%. A cautious investor with A = 6 puts only 0.0432/(6 × 0.0256) = 28.1% in the market, for 5.42% expected, 4.5% standard deviation and a certainty equivalent of 4.81%. Same menu, same beliefs, different dose, just as two-fund separation predicts.
One picture holds the geometry behind that arithmetic. An indifference curve joins the (σ, E(R)) points that give one investor the same utility, so it's E(R) = U + 0.5 × A × σ², an upward-opening parabola whose steepness is set by A. For the A = 2 investor at a certainty equivalent of 6.02%, the curve passes through 7.02% at a 10% standard deviation and 10.02% at 20%. You're at the optimum where the highest reachable curve just touches the capital market line. Check the tangency condition by hand once. An indifference curve's slope is A × σ, which at the A = 2 investor's optimum is 2 × 0.135 = 0.270, the same number as the line's Sharpe ratio. Run the same check for the A = 6 investor and you get 6 × 0.045 = 0.270 again. Every investor stops where their own steepness matches the market's price of risk.
Capital market line vs security market line
One contrast to finish, because people confuse these two lines constantly and they aren't the same object:
| Capital market line | Security market line | |
|---|---|---|
| Horizontal axis | Total risk, σ | Systematic risk, β |
| What sits on it | Efficient portfolios only | Every asset and portfolio, efficient or not |
| What it prices | The whole risk of an efficient holding | Only the undiversifiable part |
| Equation | E(R) = Rf + [(E(Rm) − Rf)/σm] × σ | E(R) = Rf + β × (E(Rm) − Rf) |
| A point below it means | The portfolio is inefficient, and diversification alone fixes it | The asset is overpriced given its beta, and only a price change fixes it |
| A point above it means | Impossible in equilibrium, so check your inputs | The asset is underpriced, which is what an active manager claims to find |
If you carry one line away, make it this one. Everything plots on or below the capital market line because most portfolios are badly diversified, while everything plots on the security market line unless it's mispriced. CAPM, coming up next, builds that second line from scratch.
Hold on to the two ideas you've just installed, because CAPM is nothing but their collision: (1) diversified investors all hold, in effect, the market portfolio, and (2) the risk any single stock contributes is therefore its co-movement with that portfolio, while its own σ drops out.