Every named formula in the series, chapter by chapter, each with its meaning in words and a link to the worked example.
Chapter 2 · Risk and return, the types
In words, the arithmetic mean adds the yearly returns and divides by how many there are.
In words, the geometric mean multiplies the yearly growth factors, takes the n-th root and subtracts one.
In words, the annualised return is one plus the total return raised to one over the number of years, minus one.
In words, the standard deviation squares each year's distance from the mean, averages those squares and takes the square root.
In words, the Sharpe ratio is the average return above the risk-free rate divided by the standard deviation.
In words, beta is the covariance of the stock with the market divided by the market's variance, which equals the correlation times the stock's standard deviation over the market's.
Chapter 3 · Probability and statistics for investors
In words, the expected value is the sum of each outcome multiplied by its probability.
In words, the variance is the probability-weighted average of each outcome's squared distance from the expected value.
In words, the geometric mean is roughly the arithmetic mean minus half the variance.
In words, portfolio variance is each asset's weighted variance plus twice the weighted covariance between them.
In words, the standard error is the yearly standard deviation divided by the square root of the number of years.
In words, the chance of k successes is the number of ways to choose which tries succeed, times the chance of k successes and n minus k failures in one such order.
In words, the chance of the event given the signal equals the chance of a true alarm divided by the chance of any alarm, true or false.
Chapter 4 · Time value of money and compounding
In words, the future value equals today's sum multiplied by one plus the rate, once for every period.
In words, the present value equals the future sum divided by one plus the rate, once for every period.
In words, the value of a level stream is the payment times one minus the discount factor for the last date, divided by the rate.
In words, a level payment forever is worth the payment divided by the rate.
In words, a payment that grows at g forever is worth the first payment divided by the rate minus the growth rate.
In words, net present value is the sum of every cash flow, each divided by one plus the discount rate once for every year until it arrives.
In words, the number of years equals the log of the target over today's sum, divided by the log of one plus the rate.
In words, one plus the nominal rate equals one plus the real rate times one plus inflation.
Chapter 5 · Microeconomics: supply, demand, elasticity, market structure, moats
In words, the price elasticity of demand is the percentage change in quantity divided by the percentage change in price.
In words, the share of the price that is markup over marginal cost equals one over the size of the demand elasticity the firm faces.
In words, the price change is the size of the shift divided by the sum of how strongly buyers and sellers respond.
In words, economic profit is the return on invested capital minus the weighted average cost of capital, times the capital invested.
In words, the moat is worth the sum of each year's shrinking excess return on capital, discounted back to today.
Chapter 6 · Macroeconomics: GDP, inflation, unemployment, the business cycle
In words, one plus real growth equals one plus nominal growth divided by one plus the rise in the price deflator.
In words, the output gap is about minus Okun's coefficient times the amount by which unemployment exceeds its natural rate.
In words, after n years a dollar buys one divided by one plus inflation to the power n of what it bought at first.
Chapter 7 · Monetary and fiscal policy, the Taylor rule and the yield curve
In words, the policy rate equals the neutral real rate plus inflation, plus half the gap between inflation and its target, plus half the output gap.
In words, the multiplier is one over one minus the share of each dollar that is spent again at home.
In words, next year's debt ratio is this year's grown by one plus the interest rate over one plus the growth rate, minus the primary surplus.
Chapter 8 · Bonds and the maths of basis points
In words, a bond's price equals every coupon and the face value added up, each one divided by one plus the period yield once for every period until it arrives.
In words, Macaulay duration is the average time until the bond's cash arrives, with each date weighted by the present value of the payment due then.
In words, modified duration is Macaulay duration divided by one plus the period yield, and it equals the percentage fall in price for each unit rise in the yield.
In words, convexity measures how fast duration itself changes as the yield moves, per dollar of price.
In words, the percentage price change is about minus duration times the yield change, plus half the convexity times the yield change squared.
In words, DV01 is the market value of the holding times its modified duration times one basis point, the dollar change for a 0.01 percentage point move in yield.
In words, one plus the nominal rate equals one plus the real rate times one plus inflation.
In words, the taxable-equivalent yield is the tax-free yield divided by one minus the investor's tax rate.
Chapter 9 · Credit: corporate bonds, munis, ratings, spreads, default and recovery
In words, expected loss equals the probability of default times the loss given default times the exposure at default.
In words, a credit spread is roughly the yearly expected loss rate plus a premium for bearing default risk and illiquidity.
In words, the breakeven default rate is the spread divided by one minus the recovery rate.
In words, the chance of no default for n years is one minus the yearly rate, multiplied by itself n times, and the cumulative default rate is one minus that.
Chapter 10 · Equity valuation
In words, the value of a share equals next year's dividend divided by the gap between the required return and the growth rate.
In words, a justified forward P/E equals the payout ratio divided by the gap between the required return and growth.
In words, a firm's value equals next year's earnings, less the share it must reinvest to grow, divided by the gap between the required return and growth.
In words, the weighted average cost of capital is the cost of equity times the equity share plus the after-tax cost of debt times the debt share.
In words, the return a buyer can expect over the long run equals the dividend yield plus the growth rate of dividends.
In words, the value of a business equals each forecast year's free cash flow discounted at the WACC, plus a terminal value for the years beyond, itself a Gordon value discounted back from the end of the forecast.
Chapter 11 · Equity risk drivers, factors, international and emerging markets
In words, a stock's expected return equals the risk-free rate plus its beta times the market's premium over the risk-free rate.
In words, beta equals the stock's correlation with the market times its volatility divided by the market's volatility.
In words, expected return equals the risk-free rate plus the stock's exposure to each factor times that factor's premium, summed over the market, size and value factors.
In words, the portfolio's volatility is the square root of the shared part of the variance, which stays, plus the company-specific part divided by the number of stocks.
In words, one plus the dollar return equals one plus the local return times one plus the currency return.
Chapter 12 · Real estate and REITs
In words, a property's value equals its net operating income divided by the cap rate.
In words, a cap rate equals the total return buyers require less the growth rate they expect for the building's income.
In words, the percentage change in the investor's equity equals the percentage change in the property's value times the value divided by the equity.
Chapter 13 · Commodities and gold
In words, the futures price equals the spot price grown at the interest rate plus the storage cost, less the benefit of holding the good, over the time to delivery.
In words, the roll yield is the gap between the expiring price and the next contract's price, as a share of the expiring price.
Chapter 14 · Currencies and what moves them
In words, the forward rate equals the spot rate times one plus the home interest rate, divided by one plus the foreign interest rate.
In words, over years the dollar price of a foreign currency should rise by the ratio of home to foreign inflation, compounded.
Chapter 15 · Forwards, futures and swaps
In words, the forward price is the spot price grown at the risk-free rate less the income the asset pays, for the time to delivery.
In words, the best hedge ratio is the correlation between the two prices times the ratio of their volatilities.
In words, the par swap rate equals one minus the final discount factor, divided by the sum of all the discount factors.
Chapter 16 · Options: European, American, Bermudan and exotics; binomial, Black–Scholes, the Greeks
In words, a call minus a put with the same strike and expiry is worth the share price minus the strike discounted at the risk-free rate.
In words, is the probability that makes the share's expected growth equal the risk-free rate.
In words, the call is worth the share price times one probability-like factor, minus the discounted strike times another.
In words, the put is worth the discounted strike times the chance of ending in the money, minus the share price times the put's share-equivalent.
In words, the time decay plus the gain from curvature plus the interest on the hedge equals the risk-free return on the option's value.
In words, the daily standard deviation the options imply is the VIX divided by the square root of 252.
Chapter 17 · Credit derivatives and structured products: CDS, MBS, ABS, CDOs, tranching and 2008
In words, the chance the borrower is still paying at time is the exponential of minus the hazard rate times .
In words, the CDS spread is about the hazard rate times the loss given default, so the hazard rate is about the spread divided by one minus the recovery rate.
In words, the premium leg equals spread times notional times the risky annuity, and the protection leg is the loss given default times the notional times the hazard rate times the same annuity.
In words, a tranche loses nothing until pool losses reach its attachment point, then loses every further dollar until its detachment point, measured as a share of its thickness.
In words, in a bad state of the economy (low ) the default rate rises above its average, and the correlation sets how far it swings with the economy.
Chapter 18 · Portfolio construction: MPT, the efficient frontier, CAPM, factors, rebalancing
In words, the portfolio's expected return is each asset's expected return times its weight, added up.
In words, the portfolio's variance is each asset's variance times its weight squared, plus twice the product of the weights, the correlation and the two standard deviations.
In words, the least-risk stock weight is the bond variance less the covariance, over the sum of both variances less twice the covariance.
In words, the tangency stock weight trades each asset's excess return against its variance and the covariance between the two.
In words, on the capital market line each unit of standard deviation earns the tangency portfolio's Sharpe ratio on top of the bill rate.
In words, the variance is one asset's variance divided by the number of assets, plus a share of the common covariance that does not shrink as assets are added.
In words, the stock share of risk is the stock weight times the stock's covariance with the portfolio, divided by the portfolio's variance.
In words, the extra growth from rebalancing is half the gap between the weighted average of the assets' variances and the portfolio's own variance.
In words, the new stock weight is the grown stock value divided by the grown value of everything.
In words, an asset's required return is the bill rate plus its beta times the market's premium over bills.
Chapter 19 · Behavioural finance
In words, a gain's felt value is the gain raised to a power below one, and a loss's felt value is minus the loss-aversion coefficient times the loss raised to a power below one.
In words, the decision weight of a chance is the chance raised to a power, divided by a scaling term that keeps the weights between zero and one.
In words, the growth needed is the required return compounded over the years, times the fall in the multiple, taken to the root of the number of years, less one.
Chapter 20 · The 11 asset classes, deepened: risk drivers and one example per driver
In words, the percentage change in price equals the old gap between return and growth divided by the new gap, minus one.
In words, one plus the dollar return equals one plus the local return times one plus the currency's change against the dollar.
In words, the taxable-equivalent yield is the muni yield divided by one minus the buyer's tax rate.
In words, a TIPS bond earns the inflation added to its principal and its real coupon, and loses its real duration times any rise in the real yield.
In words, the percentage change in equity equals the percentage change in property value divided by one minus the loan-to-value ratio.
In words, a futures investor earns the change in the spot price plus the roll yield plus the interest on the cash held as collateral.
In words, a past price in today's money equals the past price times today's CPI divided by the CPI at the time.
In words, the carry return equals the grown value of the investment, after the currency move, minus what the borrowed money costs to repay.
Chapter 21 · Fiduciary duty, advisers vs brokers, the regulators
In words, the value after years equals the starting value grown at the gross return less the fee, once for each year.
In words, the break-even holding period is the log of what the load leaves you divided by the log of the yearly fee's growth penalty.
Chapter 22 · Accounts, taxes and the investor's plan
In words, the return lost to tax each year equals the yield multiplied by the tax rate on it.
In words, the taxable-equivalent yield is the tax-free yield divided by one minus the investor's tax rate.
In words, the Roth's value is the dollar after today's tax, grown untaxed.
In words, the value of a harvest is the tax saved today less today's value of the extra tax due when the investor sells the lower-basis shares.
In words, one plus the required return equals one plus the spending rate times one plus inflation.
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