Part I · Returns and risk · Chapter 1F

Correlation, diversification and liquidity

Why diversification fails when it is needed most

1 · Correlation: moving together

Correlation measures how two return series move together over a sample. It is the Pearson coefficient of clinical statistics, computed on returns. For returns $X$ and $Y$ with standard deviations $\sigma_X$ and $\sigma_Y$, it is

where the covariance $\operatorname{Cov}(X,Y) = \mathbb{E}[(X-\mu_X)(Y-\mu_Y)]$ is the average product of the two returns' deviations from their means. Dividing by the standard deviations puts $\rho$ on a scale from −1 to +1. Two hypothetical assets with volatilities of 20% and 10% and a covariance of −0.006 have $\rho = -0.006/(0.20 \times 0.10) = -0.3$. SPX and Treasury bonds had a correlation of roughly −0.3 from 2000 to 2020, meaning they tended to diversify each other.

2 · Portfolio variance: why diversification works

If you hold a fraction $w_1$ of your capital in one asset and $w_2$ in another, with volatilities $\sigma_1$ and $\sigma_2$ and correlation $\rho$, the portfolio's variance is

Beside each asset's own risk in the first two terms, the third term holds their interaction, which takes its sign and its size from $\rho$.

Where the portfolio variance comes from

Write $a = X - \mu_X$ and $b = Y - \mu_Y$ for the two deviations from the mean. The portfolio's deviation is $w_1 a + w_2 b$, whose expected square is the portfolio's variance:

$$\begin{aligned} \sigma_p^2 &= \mathbb{E}\big[(w_1 a + w_2 b)^2\big] \\ &= w_1^2\,\mathbb{E}[a^2] + w_2^2\,\mathbb{E}[b^2] \\ &\quad + 2\,w_1 w_2\,\mathbb{E}[ab] \end{aligned}$$ (1F.3)

By the definition of variance, $\mathbb{E}[a^2] = \sigma_1^2$ and $\mathbb{E}[b^2] = \sigma_2^2$, while $\mathbb{E}[ab]$ is the covariance, which (1F.1) rewrites as $\rho\,\sigma_1\sigma_2$. Substituting these three into the last line gives (1F.2).

At $\rho = +1$ the portfolio's volatility is just the weighted average, $0.6 \times 20\% + 0.4 \times 10\% = 16\%$. Every point below that is the diversification benefit, which exists only because $\rho$ is less than 1.

The figure draws this benefit for every mix of the two assets. Drag ρ from −0.3 up to +1 and watch the gap below the dashed line close.

correlation ρ

−0.30

weight w₁

60%

σ₁

20%

σ₂

10%

σp = 11.5%

at ρ = +1: 16.0%

diversification benefit = 4.5 pts

Many assets, and why Sharpe ratios do not add

With many assets, collect the weights in a column vector $\mathbf{w}$ and the variances and covariances in a covariance matrix $\boldsymbol{\Sigma}$. Each asset's variance sits on the diagonal and each pair's covariance off it, as in the two-asset case:

For any number of assets, the portfolio variance is then

Multiplying out the two-asset case gives back (1F.2) term by term. The lesson is that individual Sharpe ratios do not aggregate: covariance dominates. Take two strategies with the same Sharpe ratio $\text{SR}$ and the same volatility, held half and half. Their mix keeps the same excess return, while (1F.2) shrinks its volatility by a factor of $\sqrt{(1+\rho)/2}$. The Sharpe ratio of the mix therefore rises by the inverse of that factor:

At $\rho = 0$, two strategies with a Sharpe ratio of 0.5 combine to $0.5 \times 1.41 = 0.71$. At $\rho = 1$ they combine to 0.5, with no gain at all. The Sharpe ratio of the mix depends on $\rho$, which neither strategy's Sharpe ratio contains.

3 · Tail correlation: when it matters

Tail correlation is correlation conditional on extreme events: the same coefficient, computed only over the periods of stress. SPX and Treasury bonds had a tail correlation closer to −0.6 in 2008, when bonds rallied sharply as stocks crashed. In 2022 it was +0.5 as both fell together: the tail correlation flips on regime.

Most "diversifying" strategies have decent unconditional correlation to equities, say 0.3 to 0.5, but a tail correlation that jumps toward 1.0 in crashes:

The genuine tail-diversifiers are rare:

• Trend following (Chapter 10D) has about 0.0 unconditional correlation but turns negative in slow drawdowns: its returns were positive in 8 of the 10 largest drawdowns of a 60/40 portfolio.
• Long Treasuries are regime-conditional, having worked from 2000 to 2020 and broken in 2022.
• Deep out-of-the-money (OTM) tail puts pay in the left tail but bleed premium in normal regimes.

This is why allocation by individual Sharpe ratio is the wrong framework. Allocation has to be by contribution to the tail behavior of the portfolio. The worked example's last row shows the cost: at a correlation of 1, diversification reduces volatility by nothing.

4 · Liquidity: the cost of getting out

Liquidity is the cost of getting into and out of a position, and it has three components:

1. The bid-ask spread, the immediate cost of executing. The bid is the highest price a buyer offers, the ask the lowest price a seller asks, and the mid-quote the point halfway between.
2. Market impact, how much your own order moves the price.
3. Stress liquidity, how spreads widen in volatile conditions.

For SPX and XSP options, normal-condition spreads are tight, a few cents on liquid strikes. In stress, spreads can widen 10 to 50 times, so the cost of unwinding a position right when you most need to is enormous. Govindaraj, Li and Zhao (2020) documented pre-earnings bid-ask spreads on single-stock options averaging 31.7% of the mid-quote.

Academic backtests use mids, while live profit and loss (P&L) uses real fills. The gap between paper and reality is mostly here (Chapter 2D and Chapter 13A).

5 · Signal, strategy and allocation

Three different things get called "an edge", and conflating them produces confused thinking:

• A signal is an observation that updates probabilities, such as the ratio of the VIX to the VIX3M rising above 1.05. The VIX measures the implied volatility of SPX options over the next 30 days and the VIX3M over the next 3 months (Chapter 9A).
• A strategy is a rule that translates signals into trades, such as "halt put-writing when the signal triggers".
• An allocation is how much capital each strategy gets, such as "10% of portfolio risk to the put-write sleeve".

On its own, a signal is not tradable. A strategy without an allocation is only a backtest. An allocation without strategies is a portfolio policy. When someone calls something "a great edge", ask whether they are describing a signal, a strategy or an allocation. Most "great edges" are signals dressed up as strategies, with no allocation discipline behind them.

6 · The six equations the book is written in

Learn the meaning, not just the form, of the six equations the book uses again and again.

1. Expected value (Chapter 1C), where $p$ is the win probability, $G$ the gain on a win, $L$ the loss on a loss and $c$ the friction of commissions, taxes and slippage:

2. The Sharpe ratio (Chapter 1D), excess return per unit of volatility, with $\mu$ the mean return, $r$ the risk-free rate and $\sigma$ the volatility:

3. Tail loss, or expected shortfall (Chapter 1B): the average loss conditional on being in the worst tail. At the 95% level it averages the worst 5% of losses, those beyond the value at risk $\text{VaR}_{95}$. In the formula below, the loss $L$ is the return with its sign reversed. This makes $\text{ES}_\alpha$ Chapter 1B's expected shortfall written as a positive number.

4. The retail-realism haircut (Chapter 13A) multiplies a paper Sharpe ratio by one factor $h$ for each layer of friction, each typically 0.7 to 0.9. The cumulative haircut is often 0.3 to 0.5: three factors of 0.75 multiply to 0.42, so a paper Sharpe ratio of 0.8 would become about 0.34.

5. Portfolio variance, (1F.6) above, is the reason individual Sharpe ratios do not aggregate: covariance dominates.

6. The conviction threshold (Chapter 13B): trade only if the estimated net return $\hat{\mu}_\text{net}$ beats a penalty $\lambda$ on tail risk plus a penalty $\kappa$ on execution cost $C_\text{exec}$.

Chapter 1G uses these primitives to explain why edges exist at all, sorting them into four sources: risk premium, behavioral, institutional and microstructure. Chapter 1D has already shown why the Sharpe ratio flatters short-volatility strategies. Chapter 13A uses the haircut equation to derive realistic retail expectations. Without these primitives those chapters would be claims, but with them they are conclusions. The rest of the book assumes the primitives of Chapters 1A to 1F are second nature.

7 · Check yourself