From the Kelly criterion to Markowitz, CAPM, value-at-risk, credit risk and the disasters that taught the industry to take all of this seriously.
Wilmott opens the portfolio chapters with the surprising claim that the best place to learn position sizing is the casino. Three takeaways:
Edward Thorp showed that a player who tracks the deck can flip the house edge in their favour by a small margin. The game isn't won with one bet — it's won by sizing many marginal-edge bets correctly.
For a binary bet that pays $b$:1 with probability $p$:
$$f^* = \frac{p(b+1) - 1}{b}.$$
$f^*$ is the fraction of wealth to risk — it maximises the expected log-growth rate.
If three bookmakers quote prices on the same outcome that, summed across all outcomes, give a payoff less than 1 in every state, take all sides. A genuine arb — no probability assumed.
The recurring lesson: edge is necessary but not sufficient. You also need to size the bet so that bad luck doesn't ruin you before good luck pays out. The mathematics of sizing is what generalises to portfolio construction.
This deck collapses Wilmott's chapters 20 through 26 into one tour. The interactive widget on slide 10 builds a two-asset efficient frontier, marks the current portfolio, and shows the VaR cut-off on the corresponding return PDF.
Variance of a weighted sum of correlated returns:
$$\mathrm{Var}\!\left(\sum_i w_i R_i\right) = \sum_i w_i^2 \sigma_i^2 + \sum_{i \ne j} w_i w_j\, \rho_{ij}\, \sigma_i \sigma_j.$$
The cross terms can be positive, negative or zero. If we mix assets that aren't perfectly correlated, the portfolio variance drops below what you'd get from any of the individual assets. The expected return is just the weighted average — no degradation. Hence "free lunch".
For a portfolio with weight $w$ in asset 1 and $1-w$ in asset 2:
$$\sigma_P^2 = w^2 \sigma_1^2 + (1-w)^2 \sigma_2^2 + 2 w (1-w) \rho \sigma_1 \sigma_2.$$
Minimum at
$$w^* = \frac{\sigma_2^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}.$$
If all $n$ assets have the same vol $\sigma$ and pairwise correlation $\rho$, an equally-weighted portfolio has
$$\sigma_P^2 = \frac{\sigma^2}{n} + \rho \sigma^2 \frac{n-1}{n}.$$
As $n \to \infty$ this tends to $\rho \sigma^2$ — you can diversify away idiosyncratic risk but never the systemic part.
The "$\rho \sigma^2$ floor" is exactly what 2008 demonstrated: every credit-sensitive name suddenly correlated near 1, and the diversification benefit that risk models had assumed evaporated.
Markowitz (1952) made the leap from descriptive ("here is what diversification does") to prescriptive ("here is the best portfolio"). For $n$ assets with expected returns $\boldsymbol\mu$ and covariance $\Sigma$, find weights $\boldsymbol w$ to
$$\min_{\boldsymbol w} \boldsymbol w^\top \Sigma \boldsymbol w \quad \text{subject to } \boldsymbol w^\top \boldsymbol\mu = \mu_P, \;\; \boldsymbol w^\top \mathbf{1} = 1.$$
The set of $(\sigma_P, \mu_P)$ pairs achievable by an efficient portfolio — one that minimises variance for its return. A hyperbola opening to the right in $(\sigma, \mu)$-space.
The leftmost point on the frontier:
$$\boldsymbol w_\text{MV} = \frac{\Sigma^{-1} \mathbf 1}{\mathbf 1^\top \Sigma^{-1} \mathbf 1}.$$
The least risky combination of the risky assets.
Introduce a risk-free asset at rate $R_f$. Draw the line from $(0, R_f)$ tangent to the frontier — the tangent point is the market portfolio:
$$\boldsymbol w_\text{tan} \propto \Sigma^{-1}(\boldsymbol\mu - R_f \mathbf 1).$$
Everything investable then lives on a single straight line in $(\sigma, \mu)$ space — the capital market line. You pick your risk tolerance, mix the risk-free asset with the tangency portfolio in the right ratio, and you're done. This is the two-fund separation theorem.
Returns are characterised entirely by mean and variance. Investors care only about those two moments. The world is normally distributed and stationary. Each of these is false; all of them are useful first approximations.
If every investor solves the Markowitz problem with the same $\boldsymbol\mu$, $\Sigma$ and $R_f$, they all hold the same tangency portfolio — which therefore has to be the market portfolio $M$. CAPM is the equilibrium statement that follows:
$$\boxed{\;\mathbb{E}[R_i] = R_f + \beta_i\, (\mathbb{E}[R_M] - R_f)\;}$$
with
$$\beta_i = \frac{\mathrm{Cov}(R_i, R_M)}{\mathrm{Var}(R_M)}.$$
CAPM lays out expected returns linearly in $\beta$. Stocks that plot above the line are "cheap" (positive alpha); below the line, "expensive". This was the first systematic way to talk about whether a stock was fairly priced relative to its risk.
Half a century of empirical work has shown CAPM is a useful but imperfect benchmark — the realised return on high-$\beta$ stocks is actually lower than CAPM predicts, while small-cap and value stocks earn unexplained premia. Fama–French added size and value factors; modern factor models pile on momentum, profitability, low-vol and more.
Even if CAPM mis-prices, every risk system on every trading floor still attributes P&L to "market beta + residual." It's the language of risk, regardless of how well the model fits.
VaR$_\alpha$ over horizon $h$ is the loss that is exceeded with probability $1 - \alpha$:
$$\Pr(L > \mathrm{VaR}_\alpha) = 1 - \alpha.$$
For $\alpha = 0.95$ and $h = 1$ day, VaR is the loss that, on average, your portfolio exceeds one trading day in twenty.
If $R \sim \mathcal{N}(\mu, \sigma^2)$ and we hold notional $V$:
$$\mathrm{VaR}_\alpha = V\, (-\mu + \sigma z_\alpha),$$
where $z_{0.95} \approx 1.645$, $z_{0.99} \approx 2.326$.
If returns are jointly normal with covariance $\Sigma$ and weights $\boldsymbol w$, the portfolio variance is
$$\sigma_P^2 = \boldsymbol w^\top \Sigma \boldsymbol w,$$
and VaR uses $\sigma_P$ in the single-asset formula.
Big banks compute VaR three ways: parametric (above), historical simulation (re-price the book under the last 250 days), and full Monte Carlo. Each gives a different number; risk committees argue about which is right.
Derivative payoffs aren't linear in the underlying. Three flavours of VaR computation, in increasing accuracy and cost:
Treat the derivative position as if it were $\Delta$ units of the underlying:
$$\Delta V \approx \Delta \cdot \Delta S.$$
VaR is then the underlying VaR scaled by $|\Delta|$. Cheap; misses convexity.
Add a second-order term:
$$\Delta V \approx \Delta \cdot \Delta S + \tfrac{1}{2} \Gamma\, (\Delta S)^2.$$
Captures the asymmetry of an option position. For at-the-money options, $\Gamma$ is what makes the loss distribution skewed.
Simulate $N$ moves of the underlying(s) over the horizon, reprice the derivative from scratch on each, and read off the 5th percentile. Slow, but no Taylor truncation.
For an at-the-money long-call position, delta-only VaR is symmetric, but the actual P&L distribution is positively skewed (limited downside, unbounded upside). Delta–gamma captures most of this; full MC gets the rest.
For derivatives, vol changes are themselves a risk factor — vega. Modern VaR systems include vol shocks as one of the simulated dimensions; for an option book the vega VaR can dwarf the delta VaR in a vol-spike scenario.
Robert Merton (1974) observed that a company's equity behaves like a call option on its assets. The equity holders get the residual after debt is paid; if assets don't cover debt, equity is worthless and the company defaults.
Let $V_t$ be the value of the firm's assets, $D$ the face value of debt due at $T$. At maturity:
$$\text{equity}(T) = \max(V_T - D, 0).$$
If $V$ follows GBM, equity is a Black–Scholes call on $V$ with strike $D$. From an observable equity price, you can back out an implied $V_0$ and asset volatility — and from those, a default probability:
$$\Pr(\text{default}) = \Pr(V_T < D) = N\!\left(-\frac{\log(V_0/D) + (\mu - \sigma^2/2)T}{\sigma \sqrt{T}}\right).$$
Despite the limitations, the Merton picture is the conceptual basis for almost every "structural" credit model in use today.
The other branch of credit modelling: reduced-form models. Don't try to explain why default happens. Just model the default time $\tau$ as the first arrival of a Poisson process with intensity $\lambda(t)$:
$$\Pr(\tau > t) = \exp\!\left(-\int_0^t \lambda(s)\, ds\right).$$
If a corporate bond pays 1 at $T$ unless default occurs first (in which case it pays a recovery $R$), the price is approximately
$$B = e^{-(r+\lambda)T} + R \cdot \text{(default contribution)}.$$
For zero recovery, $B = e^{-(r+\lambda)T}$ — default intensity adds to the discount rate.
The credit spread $s$ of a bond satisfies $s \approx \lambda (1 - R)$. A 200 bp spread with 40% recovery implies a default intensity around 3.3% per year.
Reduced-form models are the workhorse of credit-default-swap (CDS) pricing. The market quotes CDS premia, and traders back out a curve of $\lambda(t)$ — just as fixed-income desks back out a yield curve.
Structural models say what defaults; reduced-form models say when. In practice quants use both — structural for capital and stress tests (where the "why" matters), reduced-form for pricing (where the "when" is what trades).
A way to glue marginals together to form a joint distribution. The Gaussian copula:
$$C(u_1, u_2) = N_2\!\left(N^{-1}(u_1), N^{-1}(u_2); \rho\right).$$
Famously used to price CDOs pre-2008 — "the formula that killed Wall Street."
Wilmott's framework for the stress scenario your normal-world risk model doesn't cover. Don't smooth the move probabilistically; ask what happens if the market drops 20% overnight and your hedges break?
| Year | Who | What broke |
|---|---|---|
| 1994 | Orange County | Leveraged reverse-repo position on rates; Fed hikes wiped out US$1.6bn. |
| 1995 | Barings | Nick Leeson short straddle on Nikkei; Kobe earthquake spiked vol; US$1.4bn loss closed the bank. |
| 1998 | LTCM | Convergence trades on rates and credit; Russia defaulted; correlations went to 1. |
| 2008 | Sub-prime CDOs | Gaussian-copula default correlation drastically underestimated tail dependence. |
| 2012 | JPM "London Whale" | Synthetic credit positions exploded; US$6bn loss. |
Every textbook chapter on portfolio risk should end with a chapter on disasters, because what makes them disasters isn't the maths failing — it's the maths working perfectly until the assumptions it relied on stopped being true.
A two-asset portfolio. Pick the expected returns, volatilities, correlation and the weight in asset 1; the widget computes the portfolio mean, vol, 95%/99% VaR and Sharpe ratio. The left chart plots the efficient frontier in $(\sigma, \mu)$ space with the current portfolio marked. The right chart is the normal return PDF with the 95% VaR cut-off shaded.
Slide the correlation toward $-1$ and watch the frontier sag toward the y-axis — perfect anti-correlation lets you build a portfolio with zero variance. Slide $w_1$ past 1 or below 0 and you're short-selling; the portfolio leaves the no-short-sale frontier and lives on its hyperbolic extension.