Review
Reviewing the Six Pictures
A review of the whole book organized around its six master pictures — what each one says, the algebra beneath it, and a set of puzzles that test whether you can reason with the figure.
The book makes a strong claim: six pictures hold the working core of investment theory. This review is organized around those six, not the eleven chapters. For each picture there is a short statement of what it says, the algebra that defines it — set aside, for the reader who wants it — and a set of puzzles.
The puzzles matter more than the equations. Knowing the formula for a bond’s duration is like knowing the firing order of an engine’s cylinders: useful to a mechanic, unnecessary to a driver. What a driver needs is to know how the machine behaves — what it does when you turn the wheel, when it will and will not respond, and what it cannot do at all. The pictures are the driver’s knowledge. Each puzzle asks you to predict how a figure behaves when its inputs change, what it makes possible, and what it rules out. Where the answer is best found by experiment, follow the link back to the live figure and push on it yourself.
Picture 1 · Compound Growth
Chapter I · Figures 1.2, 1.6
What it says. A dollar today is worth more than a dollar tomorrow, because a dollar today can be invested and grown. Present value runs that logic backward: the worth today of a future sum is that sum discounted by the return it would otherwise earn. Compounding and discounting are one curve traced in two directions.
The algebra
$$FV = PV\,(1+r)^t \qquad\qquad PV = \frac{FV}{(1+r)^t}$$
Present value of a level income of $C$ per year, for $T$ years and forever:
$$PV = \frac{C}{r}\left[1-(1+r)^{-T}\right] \qquad PV_\infty = \frac{C}{r}$$
PuzzleAt 6 percent, a dollar due in ten years is worth about 56 cents today. Is a dollar due in twenty years worth half of that — 28 cents?
AppliedYou want to stop working in twenty years and then draw 300,000 dollars a year, indefinitely, from your savings. If your investments earn 6 percent, how large a sum invested today would fund that?
PuzzleYour investments will not earn exactly 6 percent each year; returns wander. How would you put a range around the nest egg — and why does a longer horizon make the average return more certain but the ending wealth less so?
AI ChallengeBuild your own retirement planner. With Claude Code or Julius.ai, build a small app that takes your age, current savings, monthly contribution, and target retirement income, then runs a Monte Carlo simulation drawing each year’s return from a historical mean and volatility. Have it draw the trumpet of possible ending wealth and report the probability your plan funds the income you want. Then answer a real question with it: what monthly contribution gives you an 80 percent chance of success?
Tools: Claude Code or Julius.ai · Deliverable: a working simulator, and the contribution that reaches 80 percent.
- Build a portfolio poised for future dividend growth: use AI to screen for firms with sustainable, rising payouts, and value them with the growing-perpetuity formula $P = \text{CF}/(R-g)$ — where the whole result turns on the assumed growth rate $g$.
Picture 2 · Risk & Return
Chapter II · The Efficient Frontier
What it says. Combining assets that do not move in lockstep produces portfolios with less risk than their parts, or more return for the same risk. The set of best such combinations is the efficient frontier. Diversification is the one gain in finance available without a matching cost.
The algebra
$$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,w_1 w_2\,\rho_{12}\,\sigma_1\sigma_2$$
The cross term, through the correlation $\rho_{12}$, is where diversification acts.
PuzzleTwo assets have the same expected return and the same volatility. As you blend them, when does the blend have lower risk than either one alone — and what does the frontier look like if they are perfectly correlated?
PuzzleYou hold $n$ assets, each with the same volatility $\sigma$, in equal weights. How fast does portfolio risk fall as $n$ grows — and does it fall all the way to zero?
AI ChallengeDraw a real efficient frontier. Ask an AI tool to pull ten years of monthly returns for five assets you care about — a few stocks, an index fund, gold, bonds — estimate their covariance matrix, and plot the efficient frontier. Mark where an equal-weighted portfolio of the five sits, and where your own current holdings sit. How far inside the frontier are you, and what reallocation would move you onto it?
Tools: Julius.ai or Claude Code · Deliverable: a plotted frontier with your portfolio located on it.
- Build a portfolio of ETFs designed to hold up through serious political disruption — macroeconomically hedged for the long term. Ask the AI to identify which holdings move together in a crisis, and assemble a mix whose parts do not all fall at once.
Picture 3 · The Price of Risk
Chapters IV–VI · The Security Market Line
What it says. In equilibrium every investor holds the same risky portfolio — the market — levered up or down with the riskless asset. An asset’s expected return then depends on one thing: how much it adds to the risk of that market portfolio, measured by its beta. The Security Market Line plots that relation. Risk you can diversify away earns nothing; only systematic risk is priced.
The algebra
$$E[R_i] = R_f + \beta_i\,(E[R_m]-R_f), \qquad \beta_i = \frac{\mathrm{cov}(R_i,R_m)}{\mathrm{var}(R_m)}$$
With several priced sources of risk (the Arbitrage Pricing Theory):
$$E[R_i] = R_f + \sum_k \beta_{i,k}\,\lambda_k$$
When to use whichAn adviser is building a client’s portfolio. When is the right tool the Markowitz frontier of Picture 2, and when is it the CAPM of Picture 3?
AppliedYou believe a stock is underpriced — it sits above the Security Market Line. How do you capture that mispricing without betting on which way the market goes?
PuzzleYou run that same hedged trade on many different underpriced stocks. What happens to the risk of the whole book as you add more positions?
AI ChallengeBuild a real-time indicator for the price of risk. Build an app that plots a universe of stocks in return-versus-beta space, fits the Security Market Line through them, and flags the names sitting above it — the candidates the model calls underpriced. Then extend the line to a plane: fit a multifactor model (market, size, value, momentum) and rank stocks by their distance above the fitted security-market plane, the multifactor analog of sitting above the SML. Run it on live prices and watch which names drift above the surface and below it.
Tools: Claude Code, with a market-data feed · Deliverable: a live screen ranking stocks by their distance above the SML and above the factor plane.
- Find what has driven a single stock: decompose its past returns into factors — market, size, value, momentum, and any macro or industry driver — then specify the hedge that would have isolated its stock-specific return, and check how that hedged return actually behaved.
- Build a stock-only portfolio hedged against inflation, by tilting toward names whose returns have risen with inflation — one of the canonical macro factors of the Arbitrage Pricing Theory.
Picture 4 · Market Efficiency
Chapter VIII · Figures 8.1, 8.4, 8.5, 8.6
What it says. A price in a competitive market already reflects what is known. New information moves it at once; old information cannot predict its next move, so prices wander like a random walk. The practical consequence: beating the market reliably from public information is very hard.
The algebra
The random walk has no formula so much as a signature. Over $t$ periods the variance of the cumulative return grows in proportion to $t$, so its standard deviation grows as $\sqrt{t}$ — the same square-root law as the trumpet of Picture 1. Independent increments, no memory, no predictable drift to trade on.
PuzzleIn the excess-of-market panels, industries that had boomed and industries that had crashed both settle onto the same flat path after the event. What would the picture look like if the market were not efficient — and what could you do about it?
PuzzleIn the merger event study, the cumulative return climbs steeply before the announcement and goes flat after it. Why that shape, and what does each half tell you?
PuzzleThe share of active funds that the index beats rises the longer the horizon you measure. Why should time make the index harder to beat?
AI ChallengeSearch for a merger before it is announced. Choose an industry and use AI to search recent news, filings, and market chatter for signals that raise the odds of a takeover — activist stakes, unusual options activity, strategic-review language, analyst speculation. Rank two or three plausible targets by the probability you would assign each. Then test yourself against the efficiency picture: for a merger that has already happened, pull the price history and see whether the run-up began before the public announcement — and ask what that implies about whether your search could have earned anything.
Tools: an AI research agent with web access · Deliverable: a ranked target list, and an event-study check on a past deal.
Picture 5 · The Payoff
Chapter IX · Figures 9.1, 9.3
What it says. An option’s payoff is bent — a straight line above the strike, flat below it, meeting at a kink where the right to choose becomes worth exercising. That bend gives an option its defining property: it responds to gains and losses asymmetrically, so its value rises with the volatility of whatever it is written on. The same bent payoff appears wherever a claim is contingent — including inside a firm’s own capital structure.
The algebra
$$\text{Call payoff} = \max(S-K,\,0)$$
Put–call parity, and the firm as an option:
$$C - P = S - K e^{-rT}, \qquad \text{Equity} = \text{a call on the firm’s assets struck at the face of its debt.}$$
PuzzleHold a firm’s assets fixed and raise their volatility. What happens to the value of its equity, and to its debt — and why does that set shareholders and bondholders at odds?
AppliedYou own a stock and fear it will fall, but you do not want to sell. What option position protects you, and what does the combined payoff look like?
AppliedYou buy a call struck at 50 for a premium of 4. Above what stock price do you make money, and how much can you lose?
AI ChallengeBuild an option strategy tool. Use Claude Code to build a tool that combines stock, calls, and puts and shows the resulting payoff diagram and its Black–Scholes value as you change the inputs. Use it to design a protective put for a stock you own, and to watch what rising volatility does to the value of a call. For a harder version, pick a firm with real leverage, treat its equity as a call on its assets, and estimate how much value would shift from creditors to shareholders if the firm’s asset volatility rose.
Tools: Claude Code · Deliverable: a working payoff-and-value tool, and a protective put designed with it.
Picture 6 · Bond Duration
Chapter XI · Figures 11.3, 11.4, 11.5
What it says. A bond is a fixed set of future payments, so its price is those payments discounted at the market rate. Plotted against the rate, the price falls and bows upward — a convex curve. Its slope is duration, the bond’s sensitivity to small rate moves; its curvature is convexity, which makes the bond fall less and rise more than duration alone predicts. Default risk adds a second dimension: the promised yield exceeds the yield you can expect to receive.
The algebra
$$P = \sum_t \frac{C_t}{(1+y)^t}, \qquad D = -\frac{1}{P}\frac{dP}{dy}$$
Price change for a yield move $\Delta y$, to second order:
$$\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2}\,\text{(convexity)}\,(\Delta y)^2$$
PuzzleYou hedge a bond so that a duration match cancels its response to a rate change. For a tiny move the hedge holds; for a large move it does not. Why — and does the leftover work for you or against you?
PuzzleThe yield curve inverts — the ten-year rate falls below the two-year. What is the market saying about the future, and why is this watched as a recession signal?
PuzzleA corporate bond promises an 8 percent yield while a Treasury of the same maturity yields 4 percent. Is your expected return 8 percent? If not, where does the difference go?
AI ChallengeStress-test a bond. Use an AI tool to fetch today’s Treasury yield curve and the terms of a specific corporate bond, compute its duration and convexity, and simulate its price under a one- and two-point rate shock — showing where the duration estimate and the true price diverge. Then split the bond’s yield spread over Treasuries into the part that compensates for expected default loss and the part that is a risk premium. As an extension, have the app flag whether the current curve is inverted, and what that has historically signaled.
Tools: Claude Code or Julius.ai · Deliverable: a duration-and-convexity calculator with a spread decomposition.
Key equations, gathered in one place
| Concept | Equation |
|---|---|
| Holding-period return | $R_t = (P_{t+1} - P_t + D_t)/P_t$ |
| Present value | $PV = FV/(1+r)^t$ |
| Perpetuity / annuity | $PV_\infty = C/r; \quad PV_T = (C/r)[1-(1+r)^{-T}]$ |
| Portfolio variance | $\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_{ij}$ |
| Sharpe ratio | $S = (E[R_p] - R_f)/\sigma_p$ |
| CAPM | $E[R_i] = R_f + \beta_i(E[R_m] - R_f)$ |
| Beta | $\beta_i = \mathrm{cov}(R_i, R_m)/\mathrm{var}(R_m)$ |
| APT ($K$ factors) | $E[R_i] = R_f + \sum_k \beta_{i,k}\lambda_k$ |
| Call payoff / parity | $\max(S-K,0); \quad C - P = S - Ke^{-rT}$ |
| Bond price / duration | $P = \sum_t C_t/(1+y)^t; \quad D = -\frac{1}{P}\,dP/dy$ |
| Growing perpetuity | $P = \text{CF}/(R - g)$ |