Utility and mean–variance preferences

Risk aversion as a number

Case study 2.4.5 gave an investor with $100,000 the utility $u(w) = \ln w$, a concave curve that scores each level of wealth $w$. A fair coin flip of $20,000 either way was worth $97,979.59 to them for sure: its certainty equivalent.

The gap to $100,000, $2,020.41, is the risk premium. It was close to $\frac{1}{2}A(w)s^2$, which is $2,000 here. The stake is $s$, and $A(w) = -\frac{u''(w)}{u'(w)}$ is the Arrow–Pratt coefficient.

Portfolios are measured in returns, not dollars. The flip is a return of +20% or −20%: an expected return $\mu = 0$ and a volatility $\sigma = 0.2$. Its certainty-equivalent return is $\frac{97{,}979.59}{100{,}000} - 1 = -2.02\%$, a sure loss just as good as the flip.

Divided by wealth, the premium $\frac{1}{2}A(w)s^2$ becomes $\frac{1}{2}\gamma\sigma^2$, with $\sigma = \frac{s}{w}$ and $\gamma = wA(w)$. The number $\gamma$ (gamma) is the investor's relative risk aversion. It has no units.

• For $u(w) = \ln w$, $A(w) = \frac{1}{w}$, so $\gamma = 1$ at every level of wealth.
• The power utilities $u(w) = \frac{w^{1-\gamma}}{1-\gamma}$ give any other $\gamma$, the same at every wealth. The formula divides by 0 at $\gamma = 1$, where $\ln w$ takes its place.

The larger $\gamma$, the more risk hurts. $\gamma = 0$ gives the straight line $u = w$, which ignores risk, and $\gamma = -1$ gives $\frac{w^2}{2}$, the convex, risk-seeking curve of case study 2.4.5. Models of investors often use values from 1 to 10.

Where it comes from

The rule is lesson 2.7.1's quadratic Taylor copy of $u$. Take the flip of $\pm s$ on wealth $w$, and call the premium $\pi$.

1. 1

   $u(w \pm s) \approx u(w) \pm u'(w)\,s + \frac{1}{2}u''(w)\,s^2$

   Why?
2. 2

   Expected utility $\approx u(w) + \frac{1}{2}u''(w)\,s^2$

   Why?
3. 3

   $u(w - \pi) \approx u(w) - u'(w)\,\pi$

   Why?
4. 4

   $\pi \approx \frac{1}{2}A(w)\,s^2$, so $\frac{\pi}{w} \approx \frac{\gamma}{2}\,\sigma^2$

   Why?

The same cancellation works for any outcomes: the first-order terms average to 0, and the squares average to the variance. If every outcome rises by $\mu$, the certainty equivalent rises by about $\mu$ as well, which gives the rule.

How good the copy is

For fair flips of $\pm\sigma$, so that $\mu = 0$, the exact certainty-equivalent returns and the rule compare like this:

Up to swings of about 20% the rule is close. At $\pm 50\%$ it is far off, because the powers of the step that the copy leaves out are no longer small (lesson 2.7.1).

The other signs of $\gamma$ behave as expected. The ±20% flip is worth exactly 0 at $\gamma = 0$, and +1.98% to a risk seeker with $\gamma = -1$, against the rule's +2%.

Mean–variance utility

Scoring every portfolio by the rule gives the score portfolio theory works with.

Score Level 3's three assets (case study 3.2.5). Their expected returns are $\boldsymbol{\mu} = (0.06, 0.08, 0.11)$ and their volatilities $(0.12, 0.18, 0.25)$, in annual decimals.

At $\gamma = 0$ only the expected return counts, so C wins. At $\gamma = 4$, C's variance costs $2 \times 0.0625 = 0.125$, more than its whole 11% return.

A and B score the same when B's extra return, 0.02, just pays for its extra variance, $0.0324 - 0.0144 = 0.018$. That is where $\frac{\gamma}{2} \times 0.018 = 0.02$, at $\gamma \approx 2.2222$.

• $\gamma > 0$, risk averse: variance lowers the score.
• $\gamma = 0$, risk neutral: only $\mu$ counts.
• $\gamma < 0$, risk seeking: variance raises the score.
• Large $\gamma$: the variance term dominates, so the least risky choice wins.

What it leaves out

Mean–variance utility sees only $\mu$ and $\sigma$. Here are three bets, each with $\mu = 0$ and $\sigma = 0.2$, so each scores $-0.02$ for $\gamma = 1$ and $-0.08$ for $\gamma = 4$. Their certainty-equivalent returns differ:

A concave utility minds the rare big loss most. Mean–variance utility treats all three alike, so for lopsided bets, such as selling options, work with the utility itself.

Indifference curves

Every portfolio with the same score $U = c$ lies on one curve, $\mu = c + \frac{\gamma}{2}\sigma^2$. It is called an indifference curve: the investor does not mind which point on it they get. With volatility across and expected return up, it is half a parabola that starts at height $c$ where $\sigma = 0$.

For $\gamma = 4$, A scores $c = 0.0312$. To be as good as A, an asset with B's volatility of 18% would need $\mu = 0.0312 + 2 \times 0.18^2 = 0.096$, and B offers only 0.08. With C's 25% it would need 0.1562, against C's 0.11.

The slope $\gamma\sigma$ is the extra expected return that makes one more unit of volatility acceptable. At $\sigma = 0$ it is 0: a little risk costs almost nothing, because the cost grows like $\sigma^2$. At $\sigma = 0.18$ it is 0.72 for $\gamma = 4$, so each extra point of volatility must bring 0.72 points of expected return.

A larger $\gamma$ gives steeper curves; for $\gamma = 0$ they are flat lines.

The best portfolio for a risk aversion $\gamma$

Two assets: a vertex

Put a weight $w$ in C and $1 - w$ in A. The expected return $\mu_p(w) = 0.06 + 0.05w$ is a straight line in $w$. The variance $\sigma_p^2(w)$ is case study 1.2.4's parabola, with $\sigma_A = 0.12$, $\sigma_C = 0.25$ and covariance $\Sigma_{AC} = 0.1 \times 0.12 \times 0.25 = 0.003$.

So $U(w) = \mu_p(w) - \frac{\gamma}{2}\sigma_p^2(w)$ is a line minus a parabola that opens upwards: a parabola that opens downwards. For $\gamma = 4$:

The top lies between 0.25 and 0.5. For the algebra, multiply out the variance as $\sigma_p^2(w) = aw^2 - 2bw + \sigma_A^2$, with $a = \sigma_A^2 + \sigma_C^2 - 2\Sigma_{AC} = 0.0709$ and $b = \sigma_A^2 - \Sigma_{AC} = 0.0114$. The slope is $U'(w) = (\mu_C - \mu_A) - \gamma(aw - b)$, and setting it to 0 (lesson 2.4.1) gives the best weight.

Here $\frac{b}{a} = 0.1608$ and the tilt is $\frac{0.05}{4 \times 0.0709} = 0.1763$, so $w^* = 0.3371$: about a third in C. Its score, 0.047313, is well above A's 0.0312 alone. At $\gamma = 2$ the tilt doubles and $w^* = 0.5134$; at $\gamma = 8$ it halves and $w^* = 0.2489$.

Three assets: a Lagrange multiplier

With three assets, maximise $U(\mathbf{w}) = \mathbf{w}^\top\boldsymbol{\mu} - \frac{\gamma}{2}\mathbf{w}^\top\Sigma\mathbf{w}$ over the weights $\mathbf{w}$, fully invested: $\mathbf{w}^\top\mathbf{1} = 1$. Lesson 4.5.1's condition is $\nabla U = \lambda\nabla(\mathbf{w}^\top\mathbf{1})$. The gradient of $\mathbf{w}^\top\Sigma\mathbf{w}$ is $2\Sigma\mathbf{w}$, as in case study 4.5.5, so the condition reads

Entry $i$ says $\mu_i - \gamma(\Sigma\mathbf{w})_i = \lambda$, and case study 3.2.5 showed that $(\Sigma\mathbf{w})_i$ is asset $i$'s covariance with the portfolio. So at the best portfolio, every asset's expected return minus $\gamma$ times its covariance with the portfolio is the same number $\lambda$. Otherwise moving a little money into the asset with the higher value would raise $U$.

With the budget $\mathbf{w}^\top\mathbf{1} = 1$, this is a linear system of four equations in $\mathbf{w}$ and $\lambda$ (lesson 3.3.1):

For $\gamma = 4$ the solution is $\mathbf{w} = (0.5659, 0.1582, 0.2759)$ with $\lambda = 0.02$. The portfolio expects 7.7% with a volatility of 11.93%, a score of 4.85%. Case study 3.2.5's 50/30/20 mix has almost the same volatility, 11.94%, but expects only 7.6%.

The best portfolio lies on the frontier of case study 4.5.5: no fully invested portfolio with the same expected return has less risk. In the second picture of the figure above, it is where the highest indifference curve touches the frontier.

The answer for every $\gamma$

Solving the system with letters in place of numbers gives one formula for every investor.

It is the two-asset rule again: the minimum-variance portfolio plus a tilt towards return. Here $\mathbf{w}_{\text{mv}} = (0.7465, 0.1511, 0.1023)$ and $\mathbf{z} = (-0.7227, 0.0285, 0.6942)$, so the tilt moves money from A to C.

As $\gamma$ grows the tilt vanishes, and the best portfolio slides down the frontier to the minimum-variance portfolio. A small $\gamma$ means a large tilt: below $\gamma \approx 0.9681$ the weight in A turns negative, a short sale.

Lesson 4.6.2 adds a risk-free asset. The budget constraint then disappears, and every investor holds the same mix of risky assets.

Common mistakes

1. $\gamma$ multiplies the variance, not the volatility. For B with $\gamma = 4$, $U = 0.08 - 2 \times 0.18^2 = 0.0152$, not $0.08 - 2 \times 0.18$.
2. Keep one unit and one period. In percent, B would score $8 - 2 \times 18^2 = -640$, because $\sigma^2$ is then in squared percent. Use annual decimals for $\mu$ and $\sigma$ alike.
3. The rule copies small risks. For swings of 50%, or for lopsided bets, the certainty equivalent can be far from $\mu - \frac{\gamma}{2}\sigma^2$.
4. Mind the sign of $\gamma$. Positive is risk averse, zero ignores risk, negative seeks it.
5. Higher curves are better. An indifference curve's height at $\sigma = 0$ is its certainty-equivalent return.
6. The best portfolio depends on $\gamma$. The minimum-variance portfolio is best only as $\gamma$ grows without limit. At $\gamma = 4$ the best mix holds 27.59% in C, not 10.23%.