Portfolio Management: Markowitz, CAPM, and Modern Portfolio Theory
Up until now in Wilmott’s book, we have been hedging everything. Buy a derivative, hedge with the underlying, pocket risk-free returns. Banks love it. But not everyone plays that game. Fund managers buy and sell assets trying to beat the bank rate. They take risk on purpose. Chapter 18 is about doing that intelligently.
The Basic Setup
We hold a portfolio of N assets. Each asset i has a price $S_i$, an expected return $\mu_i$, and a volatility $\sigma_i$. The returns on different assets are correlated, and we measure that with correlation $\rho_{ij}$ between assets i and j.
We assign weights $W_i$ to each asset (what fraction of our money goes into asset i). The weights add up to one, since that is all the money we have.
The expected return on the whole portfolio is the weighted average of individual returns:
$$\mu_\Pi = \sum W_i \mu_i$$
The risk (standard deviation) of the portfolio depends on volatilities AND correlations:
$$\sigma_\Pi = \sqrt{\sum_i \sum_j W_i W_j \sigma_i \sigma_j \rho_{ij}}$$
This second formula is where the magic happens.
Why Diversification Works
Here is the simplest demonstration. Take N uncorrelated assets ($\rho_{ij} = 0$ for $i \neq j$) and weight them equally ($W_i = 1/N$). The expected return stays at the average of individual returns. But the portfolio volatility becomes proportional to $1/\sqrt{N}$.
Meaning: as you add more uncorrelated assets, risk goes down while expected return stays the same. This is free lunch territory. In the real world, assets are rarely perfectly uncorrelated, but the principle holds. Diversification reduces volatility without hurting expected returns.
From this point on, Wilmott calls volatility “risk” (bad thing, avoid it) and expected return “reward” (good thing, want more). Simple enough.
Modern Portfolio Theory: The Efficient Frontier
Harry Markowitz won the Nobel Prize for formalizing this idea. Say you have five assets, each with its own risk and reward. Some are obviously bad. If asset D has the same risk as B but lower return, nobody wants D. If D has the same return as C but higher risk, again nobody wants D. You can rule it out.
But what about two assets where one has higher return AND higher risk? That becomes a personal choice. Depends on how much risk you can stomach.
Now the interesting part. When you combine two assets in a portfolio, varying the weights from 0% to 100%, the possible risk/reward combinations trace out a hyperbola. Not a straight line. A curve. And the upper part of that curve is where you want to be, because it gives you the highest return for each level of risk.
With many assets, you get a whole region of possible portfolios. The upper boundary of that region is the efficient frontier. Any rational investor should pick a portfolio that sits on this frontier. Everything below it is leaving money on the table.
Adding a Risk-Free Asset
If you can also invest in a risk-free asset earning rate r (like government bonds), things simplify beautifully. The efficient frontier becomes a straight line from the risk-free rate point, tangent to the original curved frontier. The tangent point is called the market portfolio. The line itself is the capital market line.
In the risk-neutral world (the world of derivatives pricing), everyone sits on a flat horizontal line through the risk-free rate. In the real world with real risk, you pick your spot on that sloped capital market line.
Where to Sit on the Frontier
The frontier is objective math. But your spot on it is personal. Wilmott describes several approaches.
Maximize the Sharpe slope. Draw a line from the risk-free rate to your portfolio on the frontier. The steeper that line, the better your probability of beating the risk-free rate. The slope is:
$$s = \frac{\mu_\Pi - r}{\sigma_\Pi}$$
Pick the portfolio that maximizes this slope. The catch: when you allow unrestricted borrowing (short selling the risk-free asset to buy more risky stuff), this leads to infinite leverage. Not great in practice.
Utility functions. Economists love this approach. You draw “indifference curves” on the risk/reward chart. Each curve connects portfolios you feel equally good about. Higher curves are better. Where your highest indifference curve touches the efficient frontier, that is your optimal portfolio.
The math is clean. The problem is that nobody actually knows their own utility function.
Markowitz in Practice: The Problem
The Markowitz model needs expected returns, volatilities, and correlations for every asset. With N assets, that is $N + N + N(N-1)/2$ parameters. For 500 stocks, you need over 125,000 correlation estimates.
Most of these numbers cannot be measured accurately. Volatilities are somewhat reliable. Expected returns and correlations? Much less so. And the optimization itself is computationally expensive for large portfolios.
This practical headache is exactly why CAPM was invented.
The Capital Asset Pricing Model (CAPM)
CAPM simplifies everything by relating each asset to one market index. Instead of tracking correlations between every pair of stocks, you just measure how each stock moves relative to the market.
The return on asset i is modeled as:
$$R_i = \alpha_i + \beta_i R_M + \epsilon_i$$
Three parts: a constant drift $\alpha_i$, a component that follows the market ($\beta_i R_M$), and a random noise term $\epsilon_i$ that is unique to each asset and uncorrelated with everything else.
Beta ($\beta_i$) is the key number. It measures how sensitive an asset is to the market. A beta of 1.5 means the stock moves 50% more than the market. Beta of 0.5 means it is calmer than the market.
Wilmott shows a scatter plot of Walt Disney stock returns against S&P500 returns (1985-1997). The slope of that regression line is Disney’s beta. The intercept is alpha.
Diversifiable vs. Systematic Risk
Here is the beautiful result. When you build a large portfolio with roughly equal weights:
The noise terms $\epsilon_i$ cancel out. Their contribution to portfolio risk is proportional to $1/N$ and vanishes as you add more stocks. This is diversifiable risk, and you can get rid of it for free by simply holding more assets.
What remains is the market risk, proportional to $\beta_\Pi \sigma_M$. This is systematic risk. You cannot diversify it away. It is the risk of the overall market going down, and every stock carries some of it.
CAPM needs far fewer parameters than Markowitz. For each asset, you just need alpha, beta, and the variance of its noise term. Plus the market return and volatility. That is it.
The Multi-Index Model
One index might not capture everything. You can extend CAPM to include multiple indices: a stock market index, a bond market index, a currency index. Each asset’s return becomes a combination of responses to all these indices plus its own noise.
Wilmott’s practical advice: do not use more than three or four indices. Fewer parameters means a more robust model. At the extreme end (one index per asset), you are back to the full Markowitz model with all its data problems.
Cointegration: A Different Way to Think
Both Markowitz and CAPM depend on correlation. But correlation has a fundamental weakness. Two stocks can be perfectly correlated over short timescales yet diverge over long periods. One grows, the other decays, but day to day they move together.
Cointegration asks a different question: do these stocks stay close to each other over time? This is about stationarity. A stationary time series has a constant mean and standard deviation. It does not wander away forever. Individual stock prices are not stationary (they trend up or down), but a carefully chosen linear combination of stocks might be.
Think of it this way. Flip a coin, get +1 for heads and -1 for tails. That sequence is stationary. But the running sum of those flips (your cumulative profit/loss) is not stationary. It wanders further and further from zero.
If you find that a portfolio of 15 stocks is cointegrated with the S&P500, you can use those 15 stocks to track the index. The tracking error stays bounded. Much easier than buying all 500 stocks.
You could also cointegrate two related stocks (Nike and Reebok, for example) for pairs trading. When they diverge beyond their normal range, bet on them coming back together.
The big advantage: cointegration makes fewer assumptions about the underlying data. Volatility and correlation do not appear explicitly in the framework.
Measuring How Well You Did
After picking your strategy, you need to know if you are actually any good. The ideal performance is returns that beat the risk-free rate consistently. High average returns with wild swings might just be luck (or recklessness).
Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Standard Deviation
Treynor Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Beta
The Sharpe ratio is for evaluating your entire investment. The Treynor ratio is for evaluating one part of a bigger portfolio. When a portfolio is well diversified, the two measures converge.
Wilmott shows two managers with the same total return. The “good” manager has a smooth upward curve. The “bad” manager has a roller coaster. Same endpoint, but the bad manager caused a lot more sleepless nights getting there. Performance measurement captures that difference.
The Bottom Line
Portfolio management is about accepting risk in exchange for reward. Derivatives theory says eliminate all risk. Portfolio theory says take risk wisely. Both perspectives matter. Even if you work in derivatives pricing, the models never perfectly eliminate risk. The ideas from this chapter (diversification, efficient frontiers, systematic vs. diversifiable risk) are relevant no matter what corner of finance you work in.
The math here is not as deep as Black-Scholes. But the practical difficulties are enormous. You need reliable estimates of parameters that might not even be stable over time. The cleaner your model (CAPM over Markowitz, cointegration over correlation), the more robust your results tend to be.
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