4.6.4 Practice set: portfolio theory

Utility and mean–variance preferences

Utility and mean–variance preferences (lesson 4.6.1)

0 of 8 correct

1.
Find the mean–variance utility of a portfolio with $\mu = 0.09$ and $\sigma = 0.2$ for $\gamma = 2$
2.
By the rule, an investor values a fair flip of $\pm 10\%$ like a sure loss of 1.5%. What is their $\gamma$ ?
3.
For $\gamma = 3$ , what expected return puts a portfolio with $\sigma = 0.2$ on the indifference curve with $c = 0.04$ ?
4.
Two assets: $\frac{b}{a} = 0.3$ , $a = 0.06$ , $\mu_C - \mu_A = 0.03$ and $\gamma = 5$ . Find the best weight $w^*$ in C
5.
At the best fully invested portfolio for $\gamma = 4$ , $\mu_A = 0.06$ and $(\Sigma\mathbf{w})_A = 0.0099$ . If $\mu_C = 0.11$ , find $(\Sigma\mathbf{w})_C$
6.
At what $\gamma$ do A ( $\mu = 0.06$ , $\sigma = 0.12$ ) and C ( $\mu = 0.11$ , $\sigma = 0.25$ ) get the same mean–variance utility?
7.
Returns are in percent: $\mu = 8$ and $\sigma = 18$ . Find the mean–variance utility for $\gamma = 4$ , in percent
8.
For log utility, the exact certainty-equivalent return of a fair flip of $\pm 40\%$ is $\sqrt{1.4 \times 0.6} - 1$ . Evaluate it to four decimal places

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The tangency portfolio and the Sharpe ratio

The tangency portfolio and the Sharpe ratio (lesson 4.6.2)

0 of 8 correct

1.
Find the Sharpe ratio of a portfolio with $\mu = 0.07$ and $\sigma = 0.16$ when $r_f = 0.02$
2.
Put 125% of your capital in P ( $\mu_P = 0.1$ , $\sigma_P = 0.2$ ), borrowing the rest at $r_f = 0.02$ . What is the expected return?
3.
Two uncorrelated assets have variances 0.01 and 0.04 and excess returns 0.02 and 0.04. Find the tangency weight of the first
4.
$\Sigma = \begin{pmatrix} 0.04 & 0.01 \\ 0.01 & 0.09 \end{pmatrix}$ and $\boldsymbol{\mu} - r_f\mathbf{1} = (0.03, 0.05)$ . Find the tangency weight of the first asset
5.
For the same two assets, find the tangency portfolio's Sharpe ratio (type sqrt)
6.
$\mu_T - r_f = 0.06$ , $\sigma_T = 0.2$ and $\gamma = 3$ . What fraction of capital goes into the tangency portfolio?
7.
A fund's monthly Sharpe ratio is 0.15. What is it a year? (type sqrt)
8.
$\mathbf{1}^\top\Sigma^{-1}\mathbf{1} = 80$ and the minimum-variance portfolio expects $\mu_{\text{mv}} = 0.05$ . Find $k = (\mathbf{1}^\top\Sigma^{-1}\mathbf{1})(\mu_{\text{mv}} - r_f)$ for $r_f = 0.06$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The capital asset pricing model

The capital asset pricing model (lesson 4.6.3)

0 of 8 correct

1.
An asset's covariance with the market is 0.03 and the market's variance is 0.025. Find its beta
2.
With $r_f = 0.02$ and $\mathbb{E}[R_m] = 0.07$ , what does CAPM give an asset with $\beta = 0.6$ ?
3.
A portfolio holds 30% in an asset with $\beta = 1.4$ and 70% in one with $\beta = 0.8$ . What is its beta?
4.
$\Sigma\mathbf{w}_M = (0.01, 0.02)$ and $\mathbf{w}_M^\top\Sigma\mathbf{w}_M = 0.016$ . Find the beta of the second asset
5.
An asset has $\beta = 1.25$ and volatility 0.25; the market's volatility is 0.12. What share of its variance is systematic?
6.
Against a benchmark with $\mu_B = 0.08$ , an asset has $\beta = 1.25$ and $\mu = 0.1$ . With $r_f = 0.02$ , find its alpha
7.
With $\mathbb{E}[R_m] = 0.08$ and $r_f = 0.03$ , what does CAPM give an asset with $\beta = 0.5$ ?
8.
A benchmark holds A, B and C at 50%, 30% and 20%. Against it, A's alpha is $-0.001$ and B's is $-0.006$ . Find C's alpha

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

The tangency portfolio and betas

Write two functions, with NumPy.

tangency(mu, cov, rf) returns (w, mean, vol, sharpe). w holds the tangency weights $\frac{1}{k}\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ of lesson 4.6.2, then come the portfolio's expected return, volatility and Sharpe ratio. Use np.linalg.solve, not an inverse.
betas(cov, w) returns the array of betas $\frac{\Sigma\mathbf{w}}{\mathbf{w}^\top\Sigma\mathbf{w}}$ against the portfolio w, as in lesson 4.6.3.

Both accept lists as well as arrays. Raise ValueError when the sizes do not match, or cov is not symmetric or not positive definite (lesson 3.7.1: check its smallest eigenvalue with np.linalg.eigvalsh). tangency also raises it when $k \le 0$ , and betas when the portfolio has no variance.

The tests use the three assets, the pair A and C, and the 50/30/20 benchmark. They check that no random portfolio beats your Sharpe ratio and that your betas rebuild the security market line. They also try a risk-free rate of 7%, an asymmetric matrix and the impossible correlations of case study 3.2.5.

See the tests

import numpy as np

_MU = np.array([0.06, 0.08, 0.11])
_SIGMA = np.array([0.12, 0.18, 0.25])
_R = np.array([[1.0, 0.3, 0.1], [0.3, 1.0, 0.5], [0.1, 0.5, 1.0]])
_COV = np.diag(_SIGMA) @ _R @ np.diag(_SIGMA)


def _raises(f, *args):
    try:
        f(*args)
    except ValueError:
        return True
    return False


def test_three_assets():
    """The three assets with rf = 3%: weights (0.5185, 0.1601, 0.3214)"""
    w, mean, vol, sharpe = tangency(_MU, _COV, 0.03)
    expect_close(w, [0.5184600126166481, 0.1601145467890288, 0.32142544059432315], what="weights", rel=1e-9)


def test_three_assets_numbers():
    """Its expected return is 7.93%, volatility 12.47% and Sharpe ratio 0.3951"""
    w, mean, vol, sharpe = tangency(_MU, _COV, 0.03)
    expect_close(mean, 0.07927356296549673, what="expected return", rel=1e-9)
    expect_close(vol, 0.12470023950760678, what="volatility", rel=1e-9)
    expect_close(sharpe, 0.3951360731948796, what="Sharpe ratio", rel=1e-9)


def test_two_assets_lists():
    """A and C alone, given as lists: 60.62% in A, Sharpe ratio 0.3878"""
    w, mean, vol, sharpe = tangency([0.06, 0.11], [[0.0144, 0.003], [0.003, 0.0625]], 0.03)
    expect_close(w, [0.6062291434927698, 0.39377085650723026], what="weights", rel=1e-9)
    expect_close(sharpe, 0.3878195977565348, what="Sharpe ratio", rel=1e-9)


def test_highest_sharpe():
    """No random fully invested portfolio has a higher Sharpe ratio"""
    w, mean, vol, sharpe = tangency(_MU, _COV, 0.03)
    W = np.random.default_rng(4).normal(size=(20_000, 3))
    W = W / W.sum(axis=1, keepdims=True)
    best = np.max((W @ _MU - 0.03) / np.sqrt(np.sum((W @ _COV) * W, axis=1)))
    assert best <= sharpe + 1e-12, f"a random portfolio has Sharpe ratio {best:.6f}, above yours, {sharpe:.6f}"


def test_rf_too_high():
    """rf = 7%, above the minimum-variance return of 6.81%: ValueError"""
    assert _raises(tangency, _MU, _COV, 0.07), "expected ValueError: k < 0, so there is no tangency portfolio"


def test_not_symmetric():
    """An asymmetric covariance matrix raises ValueError"""
    bad = _COV.copy()
    bad[0, 1] = 0.0648
    assert _raises(tangency, _MU, bad, 0.03), "expected ValueError for an asymmetric matrix"


def test_not_positive_definite():
    """Impossible correlations (0.9, 0.9, -0.9) raise ValueError"""
    R = np.array([[1.0, 0.9, 0.9], [0.9, 1.0, -0.9], [0.9, -0.9, 1.0]])
    assert _raises(tangency, _MU, np.diag(_SIGMA) @ R @ np.diag(_SIGMA), 0.03), "expected ValueError: this matrix has a negative eigenvalue"


def test_sizes():
    """Two expected returns with a 3 x 3 matrix raise ValueError"""
    assert _raises(tangency, [0.06, 0.08], _COV, 0.03), "expected ValueError when the sizes do not match"


def test_betas_tangency():
    """Betas against the tangency portfolio are (0.6088, 1.0147, 1.6236)"""
    w = tangency(_MU, _COV, 0.03)[0]
    expect_close(betas(_COV, w), [0.6088457621992378, 1.01474293699873, 1.6235886991979676], what="betas", rel=1e-9)


def test_security_market_line():
    """Against the tangency portfolio, rf + beta (mu_T - rf) gives back each expected return"""
    w, mean, vol, sharpe = tangency(_MU, _COV, 0.03)
    expect_close(0.03 + betas(_COV, w) * (mean - 0.03), _MU, what="rf + beta (mu_T - rf)", rel=1e-9)


def test_betas_benchmark():
    """Against the 50/30/20 mix, betas are (0.6833, 1.2244, 1.4551) and average to 1"""
    w = np.array([0.5, 0.3, 0.2])
    b = betas(_COV, w)
    expect_close(b, [0.6833099579242636, 1.2244039270687237, 1.4551192145862553], what="betas", rel=1e-9)
    expect_close(float(w @ b), 1.0, what="weighted average beta")


def test_betas_zero_portfolio():
    """Betas against a portfolio with no variance raise ValueError"""
    assert _raises(betas, _COV, [0.0, 0.0, 0.0]), "expected ValueError: the portfolio has no variance"

Utility and mean–variance preferences

The tangency portfolio and the Sharpe ratio

The capital asset pricing model

Coding exercise

Free further reading