Case study: the minimum-variance two-asset portfolio

The set-up

Put a fraction $w$ of your money in A and the rest, $1 - w$, in B. So $w = 0.75$ means 75% in A and 25% in B, and $w = 1$ means everything in A.

Risk is measured by volatility $\sigma$ (sigma): how widely a year's return typically swings. Desks write it as a decimal, so 20% is $\sigma_A = 0.20$, and $\sigma_B = 0.30$. Its square, $\sigma^2$, is the variance: $\sigma_A^2 = 0.04$ and $\sigma_B^2 = 0.09$.

How the two move together is measured by their correlation $\rho$ (rho), a number from $-1$ to 1. At 1 they move in perfect step, at $-1$ in exact opposition, and at 0 they are unrelated. Here $\rho = 0.2$: they tend to move the same way, but loosely.

Module 5.5 derives the variance of the portfolio, $\sigma_p^2$. Here, take it as given:

Check the ends first. At $w = 1$ only the first term is left, $\sigma_A^2 = 0.04$: all in A. At $w = 0$ only the second is left, $\sigma_B^2 = 0.09$: all in B.

Variance is a quadratic in the weight

Multiply out $(1 - w)^2 = 1 - 2w + w^2$ and $2w(1 - w) = 2w - 2w^2$, then collect the powers of $w$. The result is a quadratic $aw^2 + bw + c$ (lesson 1.2.2), with

With the numbers above, $a = 0.106$, $b = -0.156$ and $c = 0.09$. A table of the variance, and of the volatility (its square root), shows the shape:

The variance falls and then rises again, so the lowest point lies somewhere near $w = 0.75$. The table brackets it; the vertex will pin it down.

The parabola opens upwards whenever $a > 0$. Rewritten as $a = (\sigma_A - \sigma_B)^2 + 2(1 - \rho)\sigma_A\sigma_B$, it is a sum of two terms that cannot be negative while $\rho \le 1$. It is 0 only if $\rho = 1$ and the two volatilities are equal.

The minimum-variance weight

The vertex rule from lesson 1.2.2, $w^* = -\frac{b}{2a}$, gives the weight with the least variance:

Here $w^* = \frac{0.156}{0.212} \approx 0.7358$: about 74% in A and 26% in B. The lowest variance is $c - \frac{b^2}{4a} \approx 0.0326$, a volatility of about 0.1806. That is lower than A's 0.20 on its own.

The same weight gives the least volatility. A square root keeps the order of positive numbers, so the smallest variance has the smallest square root. Work with the variance, which is the quadratic, and take the square root at the end.

How correlation changes the answer

The lower the correlation, the more the two assets offset each other, and the lower the minimum risk.

At $\rho = 0.8$ the formula says to hold 123.5% in A. The extra 23.5% is paid for by a short sale of B: borrowing B's shares and selling them. The weight in B is $-0.2353$.

With $\sigma_A$ below $\sigma_B$, this happens exactly when $\rho$ is above $\frac{\sigma_A}{\sigma_B}$, here $\frac{0.20}{0.30} \approx 0.667$. Above that, mixing in B adds risk, and only a short sale of B reduces it.

Many funds may not sell short, so their weight must stay between 0 and 1. With the vertex past 1, the variance falls all the way from $w = 0$ to $w = 1$. So the best allowed weight is the end nearest the vertex: $w = 1$, all in A, with volatility 0.20.

Try it with the sliders. Raise $\rho$ past 0.667 and watch the lowest point leave $[0, 1]$, while the best long-only weight sticks at $w = 1$. Then lower $\sigma_B$ and see the threshold $\frac{\sigma_A}{\sigma_B}$ move.

What goes wrong on a desk

The formula is one line, but its inputs come from data and from other people's code. These are the cases that cause trouble.

• Perfect correlation. At $\rho = -1$ the formula gives $w^* = 0.6$ and a variance of 0: the two assets cancel exactly. At $\rho = 1$ it gives $w^* = 3$, also with variance 0, by holding 300% in A and $-200\%$ in B. Real estimates never reach $\pm 1$, so a zero-risk answer means the inputs need checking.
• No single answer. If $\rho = 1$ and $\sigma_A = \sigma_B$, then $a = 0$ and $b = 0$. Every weight gives the same variance, and the formula reads $\frac{0}{0}$. Code must check $a > 0$ before dividing.
• Impossible inputs. A correlation of 1.2 cannot occur, but a typo can produce one. Then $a = -0.014$, the parabola opens downwards, and the vertex is a maximum. The "variance" at $w = 2$ even comes out negative, $-0.038$.
• Units. Volatilities typed in percent, 20 and 30, give the same $w^*$, because only their ratio matters. But the variance then comes out in squared percent, 326.04. Mixing conventions is worse: A's monthly volatility, $0.20 / \sqrt{12} \approx 0.0577$, with B's annual 0.30 gives $w^* = 1.0015$ instead of 0.7358.
• Estimates move. Volatilities and correlations are estimated from past returns, and they change. The table shows $w^*$ moving from 0.63 to 1.24 as $\rho$ goes from $-0.5$ to 0.8. Treat $w^*$ as a guide, not a precise answer.

With many assets, the weights become a vector and the parabola becomes a bowl in many dimensions. Module 4.5 finds its lowest point.

A line in finance: the security market line

Lines from lesson 1.2.1 appear too. The capital asset pricing model (module 4.6) says a stock's expected return is a straight line in its beta $\beta$. Beta measures how strongly the stock moves with the whole market: 1 means in step with it, 2 twice as strongly.

Here $\mathbb{E}$ means "expected value", so $\mathbb{E}[R]$ is the stock's expected yearly return and $\mathbb{E}[R_m]$ the market's. $r_f$ is the risk-free rate, the return on a safe government bill.

The intercept is $r_f$: the expected return at $\beta = 0$. The slope is the market's extra return over $r_f$, in return per unit of beta. With $r_f = 3\%$ and $\mathbb{E}[R_m] = 8\%$, the line is $0.03 + 0.05\beta$.

• A stock with $\beta = 1.4$ should return 10%, one with $\beta = 0.8$ should return 7%, and one with $\beta = 1$ returns the market's 8%.
• A negative beta is allowed. A stock with $\beta = -0.5$ tends to rise when the market falls, which works like insurance. The line asks only 0.5% of it, below the risk-free rate.

A stock's distance above or below the line is its alpha. A stock with $\beta = 0.8$ expected to return 6% sits 1 point below the 7% the line asks: an alpha of $-1\%$. Going the other way, $0.03 + 0.05\beta = 0.09$ gives $\beta = 1.2$, the beta that a 9% expected return implies.

import numpy as np


def test_all_in_one_asset():
    """With all the money in A, the volatility is A's own, 0.20"""
    expect_close(portfolio_vol(1.0, 0.20, 0.30, 0.2), 0.20, what="portfolio_vol(1, 0.20, 0.30, 0.2)")


def test_vol_array():
    """portfolio_vol works on an array of weights"""
    w = np.array([0.0, 0.5, 1.0])
    expected = np.sqrt(w**2 * 0.04 + (1 - w)**2 * 0.09 + 2 * w * (1 - w) * 0.2 * 0.06)
    expect_close(portfolio_vol(w, 0.20, 0.30, 0.2), expected, what="portfolio_vol at w = 0, 0.5, 1")


def test_equal_assets():
    """Equal volatilities and zero correlation: split evenly"""
    expect_close(min_variance_weight(0.25, 0.25, 0.0), 0.5, what="min_variance_weight(0.25, 0.25, 0)")


def test_matches_brute_force():
    """The weight matches a brute-force search over 100,001 weights"""
    w = np.linspace(0, 1, 100_001)
    best = w[np.argmin(w**2 * 0.04 + (1 - w)**2 * 0.09 + 2 * w * (1 - w) * 0.2 * 0.06)]
    expect_close(min_variance_weight(0.20, 0.30, 0.2), best, what="min_variance_weight(0.20, 0.30, 0.2)", rel=1e-4)


def test_negative_correlation():
    """Negative correlation: the minimum risk is far below either asset's"""
    w = min_variance_weight(0.20, 0.30, -0.5)
    vol = portfolio_vol(w, 0.20, 0.30, -0.5)
    assert vol < 0.15, f"expected a minimum volatility below 0.15, got {vol}"


def _raises_value_error(fn, *args):
    try:
        fn(*args)
    except ValueError:
        return True
    return False


def test_impossible_correlation():
    """A correlation above 1 raises ValueError"""
    assert _raises_value_error(min_variance_weight, 0.20, 0.30, 1.2), "min_variance_weight(0.20, 0.30, 1.2) should raise ValueError"


def test_no_single_best_weight():
    """Equal volatilities with correlation 1 raise ValueError: every weight gives the same variance"""
    assert _raises_value_error(min_variance_weight, 0.25, 0.25, 1.0), "min_variance_weight(0.25, 0.25, 1.0) should raise ValueError"


def test_long_only_inside():
    """Long-only, ρ = 0.2: the vertex is already between 0 and 1, so it is kept"""
    expect_close(long_only_weight(0.20, 0.30, 0.2), 0.156 / 0.212, what="long_only_weight(0.20, 0.30, 0.2)")


def test_long_only_clips_high():
    """Long-only, ρ = 0.8: the vertex is past 1, so the answer is 1 (all in A)"""
    expect_close(long_only_weight(0.20, 0.30, 0.8), 1.0, what="long_only_weight(0.20, 0.30, 0.8)")


def test_long_only_clips_low():
    """Long-only, with A the riskier asset and ρ = 0.8: the vertex is below 0, so the answer is 0"""
    expect_close(long_only_weight(0.30, 0.20, 0.8), 0.0, what="long_only_weight(0.30, 0.20, 0.8)")