4.6.2 The tangency portfolio and the Sharpe ratio

A risk-free asset

A risk-free asset pays a return $r_f$ known in advance, such as a government bill held to maturity (case study 1.2.4). Its volatility is 0, and it has no covariance with anything. Here $r_f = 0.03$ a year, a simple return like the assets' $\boldsymbol{\mu}$ .

Take a risky portfolio P, with expected return $\mu_P$ and volatility $\sigma_P$ . Put a fraction $t$ of your money in P and the rest, $1 - t$ , in the bill.

The expected return is the weighted average $(1 - t)r_f + t\mu_P$ . For the variance, add the bill to case study 3.2.5's covariance matrix as a row and column of zeros: $\mathbf{w}^\top\Sigma\mathbf{w}$ then leaves only $t^2\sigma_P^2$ .

Rule: Mixing with the risk-free asset

\begin{gathered} \mu = r_f + t\,(\mu_P - r_f) \\[4pt] \sigma = \lvert t \rvert\,\sigma_P \end{gathered}

For $t \ge 0$ both are straight lines in $t$ , so the mixes lie on a straight line from the bill, at $(0, r_f)$ , through P. A $t$ above 1 means borrowing at $r_f$ to hold more than 100% in P.

Take P to be asset C, with $\mu_C = 0.11$ and $\sigma_C = 0.25$ :

$t$	$-0.5$	0	0.5	1	1.5
$\mu$	$-0.01$	0.03	0.07	0.11	0.15
$\sigma$	0.125	0	0.125	0.25	0.375

Half in C and half in the bill expects 7% with 12.5% volatility. With $t = 1.5$ the investor borrows half their capital at 3%: leverage, more return and more risk.

With $t = -0.5$ they sell C short and put the proceeds in the bill. The volatility is 12.5% again, but the expected return is $-1\%$ : the line folds back downwards.

t

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The Sharpe ratio

The slope of that line is the extra expected return each unit of volatility brings. Funds report it widely.

Rule: Sharpe ratio

S_P = \frac{\mu_P - r_f}{\sigma_P}

The excess return $\mu_P - r_f$ , earned for taking risk, per unit of volatility. With annual $\mu_P$ , $r_f$ and $\sigma_P$ it is an annual ratio of two returns, so it has no units.

With $r_f = 0.03$ , asset A has $S = \frac{0.03}{0.12} = 0.25$ , B has 0.2778 and C has 0.32. Case study 3.2.5's 50/30/20 mix has 0.3852, and the minimum-variance portfolio of case study 4.5.5 has 0.3476.

Leverage does not change it. For $t > 0$ , every mix of P with the bill has P's Sharpe ratio. The excess return and the volatility both scale by $t$ ; for $t < 0$ the ratio flips sign.
Signs. $S_P$ is negative when $\mu_P < r_f$ : the portfolio pays less than the bill and takes risk too. It is 0 when $\mu_P = r_f$ , and the bill itself, with $\sigma = 0$ , has none.
The period matters. Over a month the mean is about $\frac{1}{12}$ of the annual one and the volatility $\frac{1}{\sqrt{12}}$ (case study 3.2.5). So a monthly Sharpe ratio is the annual one divided by $\sqrt{12}$ : C's 0.32 a year is 0.0924 a month.

At any volatility, a steeper line from the bill reaches a higher expected return. So the best risky portfolio to mix with the bill is the one with the highest Sharpe ratio.

Each asset mixed with the bill traces a line from 3% through the asset, with its Sharpe ratio as slope. The steepest line that still reaches a portfolio touches the frontier at the tangency portfolio.

t = 0.5

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]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf = 0.03 def sharpe(w): return (w @ mu - rf) / np.sqrt(w @ Sigma @ w) print(np.round((mu - rf) / sigma, 4)) print(round(sharpe(np.array([0.5, 0.3, 0.2])), 4)) w_mv = np.linalg.solve(Sigma, np.ones(3)) w_mv = w_mv / w_mv.sum() print(round(sharpe(w_mv), 4)) for t in [0.5, 2]: m, s = rf + t * (0.11 - rf), t * 0.25 print(t, round((m - rf) / s, 4)) print(round((0.11 - rf) / 12 / (0.25 / np.sqrt(12)), 4), round(0.32 / np.sqrt(12), 4))

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The tangency portfolio

Two assets: a table

Mix A and C, with a weight $w$ in C, and compute the Sharpe ratio of each mix:

$w$	0	0.2	0.4	0.6	0.8	1
Sharpe ratio	0.25	0.3553	0.3878	0.3704	0.3435	0.32

The ratio climbs above both assets' and peaks near $w = 0.4$ . The excess return moves in a straight line between the two assets. The volatility of a mix stays below the straight line between theirs: diversification (case study 1.2.4).

The algebra: the best investor with a bill

Lesson 4.6.1's investor can now hold the bill too.

Write $\mathbf{x}$ for the amounts in the risky assets, as fractions of capital. They need not add up to 1, because the bill holds the rest, $1 - \mathbf{x}^\top\mathbf{1}$ . The expected return is $r_f + \mathbf{x}^\top(\boldsymbol{\mu} - r_f\mathbf{1})$ and the variance $\mathbf{x}^\top\Sigma\mathbf{x}$ , so

\begin{aligned} U(\mathbf{x}) = {} & r_f + \mathbf{x}^\top(\boldsymbol{\mu} - r_f\mathbf{1}) \\ & - \frac{\gamma}{2}\,\mathbf{x}^\top\Sigma\mathbf{x} \end{aligned}

There is no constraint left. Maximising $U$ is minimising $-U$ , which is lesson 4.4.1's quadratic $\frac{1}{2}\mathbf{x}^\top A\mathbf{x} - \mathbf{b}^\top\mathbf{x}$ , plus a constant, with $A = \gamma\Sigma$ and $\mathbf{b} = \boldsymbol{\mu} - r_f\mathbf{1}$ .

Its minimum solves $A\mathbf{x} = \mathbf{b}$ , so $\mathbf{x} = \frac{1}{\gamma}\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ . It is the one best point, because $\Sigma$ is positive definite (lesson 3.7.1).

The direction $\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ is the same for every $\gamma$ : only the amount changes. Scaled so its entries add up to 1, it is a portfolio of risky assets alone.

Rule: Tangency portfolio

\begin{gathered} \mathbf{w}_T = \frac{\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})}{k} \\[4pt] k = \mathbf{1}^\top\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1}) \end{gathered}

The risky portfolio with the highest Sharpe ratio, where the line from the bill touches the frontier. Its Sharpe ratio $S_T$ has $S_T^2 = (\boldsymbol{\mu} - r_f\mathbf{1})^\top\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ . It needs $k > 0$ .

For A and C, $\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1}) = (1.835, 1.1919)$ , which adds up to 3.0269. So $\mathbf{w}_T = (0.6062, 0.3938)$ : 39.38% in C, with a Sharpe ratio of 0.3878, the peak of the table.

For all three assets, $\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1}) = (1.6428, 0.5074, 1.0185)$ and $k = 3.1687$ :

\mathbf{w}_T = (0.5185, 0.1601, 0.3214)

It expects 7.93% with a volatility of 12.47%, a Sharpe ratio of 0.3951, and $\sqrt{(\boldsymbol{\mu} - r_f\mathbf{1})^\top\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})}$ gives the same 0.3951. Among 100,000 random portfolios, none does better.

\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})

import numpy as np rf = 0.03 mu2 = np.array([0.06, 0.11]) S2 = np.array([[0.0144, 0.003], [0.003, 0.0625]]) for w in [0, 0.2, 0.4, 0.6, 0.8, 1]: x = np.array([1 - w, w]) print(w, round((x @ mu2 - rf) / np.sqrt(x @ S2 @ x), 4)) raw = np.linalg.solve(S2, mu2 - rf) w_T = raw / raw.sum() print(np.round(raw, 4), round(raw.sum(), 4), np.round(w_T, 4), round((w_T @ mu2 - rf) / np.sqrt(w_T @ S2 @ w_T), 4))

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\mu_T

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]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) raw = np.linalg.solve(Sigma, mu - rf * ones) k = raw.sum() w_T = raw / k mu_T, sigma_T = w_T @ mu, np.sqrt(w_T @ Sigma @ w_T) S_T = (mu_T - rf) / sigma_T print(np.round(raw, 4), round(k, 4), np.round(w_T, 4)) print(round(mu_T, 4), round(sigma_T, 4), round(S_T, 4), round(np.sqrt((mu - rf) @ raw), 4)) rng = np.random.default_rng(1) W = rng.normal(size=(100_000, 3)) W = W / W.sum(axis=1, keepdims=True) S = (W @ mu - rf) / np.sqrt(np.sum((W @ Sigma) * W, axis=1)) print(round(S.max(), 4), S.max() <= S_T + 1e-12)

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When there is no tangency portfolio

The scaling divides by $k$ . In case study 4.5.5's letters, $k = k_2 - r_fk_1 = k_1(\mu_{\text{mv}} - r_f)$ .

Here $k_1 = \mathbf{1}^\top\Sigma^{-1}\mathbf{1} = 83.0812$ , and $\mu_{\text{mv}} = \frac{k_2}{k_1} = 6.81\%$ is the minimum-variance portfolio's expected return. So $k > 0$ exactly when the risk-free rate is below $\mu_{\text{mv}}$ .

$r_f$	$k$	$\mathbf{w}_T$	$\mu_T$	$\sigma_T$	$S_T$
0	5.6611	$(0.6189, 0.1562, 0.225)$	0.0744	0.1146	0.6489
0.02	3.9995	$(0.5658, 0.1582, 0.2759)$	0.077	0.1193	0.4773
0.03	3.1687	$(0.5185, 0.1601, 0.3214)$	0.0793	0.1247	0.3951
0.05	1.5071	$(0.267, 0.17, 0.563)$	0.0915	0.166	0.2502
0.06	0.6763	$(-0.3222, 0.1933, 1.1289)$	0.1203	0.2986	0.202
0.07	$-0.1546$	$(5.4225, -0.0335, -4.389)$	$-0.1601$	1.2202	$-0.1886$

As $r_f$ rises towards 6.81%, the tangency portfolio climbs the frontier and runs away, with ever larger positions.

Above it, no line from the bill touches the upper half of the frontier. The formula still returns weights, but they sit on the lower half and have a negative Sharpe ratio: the worst line, not the best. Code must check $k > 0$ .

At $r_f = 0.02$ the tangency portfolio is almost lesson 4.6.1's best portfolio for $\gamma = 4$ , $(0.5659, 0.1582, 0.2759)$ . That portfolio's multiplier was $\lambda = 0.02$ to four places, and with $r_f = \lambda$ exactly the two coincide. Every portfolio on the upper frontier is the tangency portfolio for some risk-free rate.

k_1 = \mathbf{1}^\top\Sigma^{-1}\mathbf{1}

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]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) ones = np.ones(3) for rf in [0.0, 0.02, 0.03, 0.05, 0.06, 0.07]: raw = np.linalg.solve(Sigma, mu - rf * ones) w = raw / raw.sum() m, s = w @ mu, np.sqrt(w @ Sigma @ w) print(rf, round(raw.sum(), 4), np.round(w, 4), round(m, 4), round(s, 4), round((m - rf) / s, 4)) x1 = np.linalg.solve(Sigma, ones) print(round(x1.sum(), 4), round(x1 @ mu / x1.sum(), 4))

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The capital market line and two-fund separation

The line from the bill through the tangency portfolio is the capital market line. Every point on it mixes $\mathbf{w}_T$ with the bill. Away from T, each point beats every frontier portfolio of the same volatility.

Rule: Capital market line

\begin{gathered} \mu = r_f + S_T\,\sigma \\[4pt] t = \frac{\mu_T - r_f}{\gamma\,\sigma_T^2} \end{gathered}

The best portfolios when a risk-free asset exists. An investor with risk aversion $\gamma$ puts the fraction $t$ in the tangency portfolio and $1 - t$ in the bill. Also $t = \frac{k}{\gamma}$ , the sum of the entries of $\mathbf{x}$ .

Here $\mu = 0.03 + 0.3951\,\sigma$ , and three investors split their money like this:

$\gamma$	in $\mathbf{w}_T$	in the bill	$\mu$	$\sigma$	$U$
2	1.5843	$-0.5843$	0.1081	0.1976	0.069
4	0.7922	0.2078	0.069	0.0988	0.0495
8	0.3961	0.6039	0.0495	0.0494	0.0398

The investor with $\gamma = 2$ borrows 58.43% of their capital at 3%. The one with $\gamma = 4$ scored 4.85% in lesson 4.6.1 without a bill, and scores 4.95% with it. Cutting risk with the bill costs less than moving down the frontier.

Two-fund separation: every investor holds the same two funds, the bill and $\mathbf{w}_T$ . Only the split depends on $\gamma$ . Case study 4.5.5's two-fund theorem built the frontier from two risky portfolios; with a risk-free asset, one of the two funds is the bill.

Try it below. Move the risk-free rate from 0 to 6% and watch the tangency portfolio climb the frontier while its Sharpe ratio falls.

Push it past 6.81%, the minimum-variance return, and the tangency portfolio disappears. The explorer finds it with short sales allowed, as the formula does.

Explore

Every mix of the three assets is a point: its volatility across, its expected return up. Move the target return and read the cheapest mix in risk that earns it. Turn on a risk-free rate to add the tangency portfolio.

Grey dots are the assets, the open circle the minimum-variance portfolio, the square the tangency portfolio and the dashed line the capital market line.

Target expected return = 0.09

Allow short salesAdd a risk-free rate

Risk-free rate = 0.03

Frontier portfolio: A 29.9%, B 16.9%, C 53.2%
Return and risk: expected return 9%, volatility 15.99%
Minimum variance: A 74.7%, B 15.1%, C 10.2%: return 6.81%, volatility 10.97%
Tangency portfolio: A 51.8%, B 16%, C 32.1%: return 7.93%, volatility 12.47%, Sharpe ratio 0.395

In practice, borrowing costs more than lending. Then lenders use the tangency portfolio for the lending rate, and borrowers a riskier one, further up the frontier, for the higher borrowing rate. Investors in between hold frontier portfolios and neither lend nor borrow.

\gamma

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]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) raw = np.linalg.solve(Sigma, mu - rf * ones) w_T = raw / raw.sum() mu_T, var_T = w_T @ mu, w_T @ Sigma @ w_T for gamma in [2, 4, 8]: x = raw / gamma t = x.sum() m, s = rf + x @ (mu - rf), np.sqrt(x @ Sigma @ x) print(gamma, round(t, 4), round(1 - t, 4), round(m, 4), round(s, 4), round(m - gamma / 2 * s**2, 4), round((mu_T - rf) / (gamma * var_T), 4))

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Lesson 4.6.3 asks what happens if every investor does this: the tangency portfolio becomes the market, and expected returns line up with beta.

Common mistakes

Subtract the risk-free rate. C's Sharpe ratio is $\frac{0.11 - 0.03}{0.25} = 0.32$ , not $\frac{0.11}{0.25} = 0.44$ .
Keep the periods together. Divide an annual Sharpe ratio by $\sqrt{12}$ for a monthly one. A monthly mean minus an annual $r_f$ is meaningless.
Leverage does not raise the Sharpe ratio. It moves along the line; only a better risky mix makes the line steeper.
Check $k > 0$ before scaling. With $r_f$ above the minimum-variance return, the formula returns the worst portfolio, not the best.
Use the whole matrix. The tangency weights are $\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ , scaled. Ranking assets by their own Sharpe ratios ignores the correlations: A has the lowest ratio, 0.25, yet the largest weight, 51.85%.
A negative amount in the bill is borrowing, and a negative weight in an asset is a short sale.

The tangency portfolio and the Sharpe ratio

0 of 12 correct

1.
Put 60% in C ( $\mu = 0.11$ , $\sigma = 0.25$ ) and 40% in a bill at $r_f = 0.03$ . What is the expected return?
2.
What is the volatility of that mix?
3.
Find the Sharpe ratio of a portfolio with $\mu = 0.09$ and $\sigma = 0.15$ when $r_f = 0.03$
4.
A portfolio has an annual Sharpe ratio of 0.4. What is its monthly Sharpe ratio? (type sqrt)
5.
That portfolio is held at 150%, borrowing 50% at $r_f$ . What is the Sharpe ratio of the position?
6.
A portfolio expects 2% when $r_f = 0.03$ , with volatility 0.1. Find its Sharpe ratio
7.
Two uncorrelated assets have variances 0.04 and 0.09 and excess returns $\boldsymbol{\mu} - r_f\mathbf{1} = (0.04, 0.06)$ . Find the tangency weight of the first
8.
For the same two assets, find the tangency portfolio's Sharpe ratio (type sqrt)
9.
The capital market line has $r_f = 0.03$ and slope 0.4. What expected return does it give at $\sigma = 0.1$ ?
10.
$\mu_T - r_f = 0.05$ , $\sigma_T = 0.125$ and $\gamma = 4$ . What fraction of capital goes into the tangency portfolio?
11.
For the same tangency portfolio and $\gamma = 2$ , what weight goes in the bill?
12.
The minimum-variance portfolio expects 6.81%. Is there a tangency portfolio when $r_f = 7\%$ ? Type 1 for yes, 0 for no

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\mu = r_f + t(\mu_P - r_f)

The tangency portfolio and the Sharpe ratio

A risk-free asset

Rule: Mixing with the risk-free asset

The Sharpe ratio

Rule: Sharpe ratio

The tangency portfolio

Two assets: a table

The algebra: the best investor with a bill

Rule: Tangency portfolio

When there is no tangency portfolio

The capital market line and two-fund separation

Rule: Capital market line

Common mistakes

The tangency portfolio and the Sharpe ratio

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