Variance and risk

Idea: Same mean, different risk

A sure $1$ and a flip that wins $3$ or loses $1$ have the same mean, but not the same risk.

• Variance: the average squared distance from the mean
• Standard deviation: its square root, in payoff units

Formula: Variance

$\mathrm{Var}(X)=E[X^2]-E[X{]}^2$

The mean of the square minus the square of the mean.

Example: Variance of the flip

A fair flip wins $3$ or loses $1$. What is the variance?

1. Mean: $\frac{1}{2}\times 3-\frac{1}{2}\times 1=1$
2. $\frac{1}{2}\times (3-1{)}^2+\frac{1}{2}\times (-1-1{)}^2=4$

Formula: Variance of a bet

$\mathrm{Var}=p(1-p)(w+l{)}^2$

Win w with chance p, else lose l. Only the gap w + l matters.

Idea: A fair die

For a fair die the variance is $\frac{91}{6}-{\left(\frac{7}{2}\right)}^2=\frac{35}{12}$.

• $n$ faces: $\frac{n^2-1}{12}$
• $8$ faces: $\frac{8^2-1}{12}=\frac{21}{4}$
• Standard deviation of a die: about $1.71$

Idea: Scaling versus repeating

Scale a bet by k: the SD grows k times, the variance k² times. n independent copies: variance n times.

• $100$ times the size: SD $200$
• $100$ independent flips: SD $20$
• Same profit, $\frac{200}{20}=10$ times less risk

Formula: Variance of a sum

$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Cov}(X,Y)$

Variances add only for independent or uncorrelated bets.

Common mistake: Double size is not 2 bets

Doubling the stake is not the same as playing on $2$ independent occasions.

Instead Variance $2^2\times 4=16$ versus $2\times 4=8$. Add variances, then take the root.

Quick check: Win 5 or lose 1

A bet wins $5$ with probability $\frac{1}{4}$ and otherwise loses $1$. What is its variance?

Idea: Say it out loud

Mean first, then the variance, then the standard deviation in the payoff's units.

• Say you assume independence before adding
• A bigger stake scales risk one for one; more bets by the square root