CFA Level 1 Cheat Sheet

Module 1.1: Rates of Return

• Total (holding period) return is $(P_1-P_0+D)/P_0$, the price return plus the capital distribution return, with each component divided by the beginning price $P_0$.
• An ex post (actual) return uses prices and income already observed, while an ex ante (expected) return is an estimate for a future period.
• The geometric mean return $[(1+R_1)(1+R_2)\cdots (1+R_n){]}^{1/n}-1$ is the measure for an annualized return or a compound annual growth rate; it cannot exceed the arithmetic mean, and the two are equal only when every period's return is identical.
• A holding period return is annualized with $(1+HPR{)}^{365/\text{days}}-1$, or $(1+R_{\text{period}}{)}^c-1$ with $c$ periods in a year; for a period shorter than a year this assumes the same return is earned in every remaining period.
• For a given positive stated annual rate, more frequent compounding gives a higher effective annual rate, $(1+r_s/m{)}^m-1$, and continuous compounding gives the highest, $e^r-1$.
• The continuously compounded return is $r_{cc}=\ln (1+HPR)$, which reduces to $\ln (P_1/P_0)$ only when no distributions are paid, and log returns are additive across periods.

Module 1.2: Components and Measures of Return

• An interest rate can be read as a required rate of return, a discount rate or an opportunity cost of current consumption; a measure of risk is not one of these interpretations.
• Required return = real risk-free rate + expected inflation + risk premia, where the real risk-free rate plus expected inflation is the nominal risk-free rate, such as a short-term government bill yield.
• The real risk-free rate reflects only time preference and is linked to the nominal rate exactly by $(1+r_{\text{nominal}})=(1+r_{\text{real}})(1+{\pi }^e)$, or approximately by $r_{\text{real}}\approx r_{\text{nominal}}-{\pi }^e$.
• Gross return deducts trading commissions and other costs of generating the return but not management and administration fees; net return deducts both.
• With one tax rate on capital gains and another on distributions, the after-tax nominal return is price return $\times (1-t_{cg})$ + distribution return $\times (1-t_d)$.
• The leveraged return is $r_L=r_P+(V_D/V_E)(r_P-r_D)$, a gain or loss measured against the investor's own cash $V_E$.

Module 2.1: Financial Assets, Instruments, and Indicators

• Total return is the capital appreciation (price) return plus the capital distribution return; distributions are realized when received, price gains only when the asset is sold, and unrealized gains still count in total return.
• Equity returns come mainly from price return, with variable and discretionary dividends, while a traditional bond's return comes mainly from its fixed coupon and price return is usually the smaller part.
• Interest rates, market indexes and foreign exchange rates are financial indicators: they influence returns but produce no cash flows of their own.
• A T-bill's holding period return is $(\text{Face value}-\text{Price})/\text{Price}$, annualized as $(1+HPR{)}^{365/t}-1$; when market interest rates rise, fixed-income prices fall, and longer-dated bonds fall more.
• The domestic currency return is $(1+r_{\text{local}})(1+r_{FX})-1$, where $r_{FX}$ is the change in the value of the foreign currency and is negative if it depreciates.
• Volatility lowers the geometric (compound) return, so the arithmetic mean overstates multiperiod performance, most of all for the more volatile asset.

Module 2.2: Long-Term Returns on Financial Assets

• Under the risk-return tradeoff, expected long-term returns in a developed market rank short-term government bills < government bonds < corporate bonds < large-cap equities < small-cap equities, with bonds compared at similar maturities.
• The credit spread is the extra return corporate bonds offer over government bonds for their greater default risk, and longer-term fixed-income securities are expected to earn more than shorter-term ones because they face more inflation and interest rate risk.
• Systematic (macro) risk, such as inflation, interest rate and market risk, affects all investments to some degree and is not reduced by diversification; unsystematic (micro) risk, such as issuer and sector risk, is.
• The range, the highest return minus the lowest return, is the simplest measure of risk, and a wider range means more variability and more risk.
• A client who needs capital growth and tolerates risk tilts toward equities, a client who needs steady income with low risk tolerance tilts toward fixed-income securities, and a client who wants both holds a balanced mix.
• Each component of a portfolio's expected return is the weighted average of the asset-class components, $E(R_p)=\sum w_{i}E(R_i)$, and total return is the capital appreciation return plus the capital distribution return.

Module 3.1: Time-Weighted and Money-Weighted Returns

• The money-weighted return is the IRR of the portfolio's cash flows, the rate that gives them an NPV of zero, with money going in and money coming out entered with opposite signs.
• The time-weighted return geometrically links the holding period returns of subperiods formed at each significant deposit or withdrawal, and is annualized with a root equal to the number of years, not the number of subperiods.
• The MWR depends on the timing and size of external cash flows: a deposit just before strong performance makes it higher than the TWR and a deposit just before poor performance makes it lower, while the TWR is unaffected.
• The MWR and the TWR are equal when there are no external cash flows during the period.
• The investment management industry prefers the TWR because managers usually do not control when clients add or withdraw money; the MWR suits a manager or investor who controls the cash flows.
• Firms claiming compliance with the voluntary GIPS must present time-weighted returns, except where the manager controls external cash flows and the portfolio is also closed-end, has a fixed life, calls committed capital or holds mainly illiquid investments.

Module 3.2: Security Market Indexes

• A price index uses constituent prices only and gives the price return; a total return index adds income, assumed reinvested, and its total return exceeds the price return when constituents pay distributions.
• A price-weighted index is the sum of prices divided by a divisor that is changed for splits, reverse splits, stock dividends and constituent changes, and its higher-priced stocks carry more weight.
• An equal-weighted index gives each constituent $1/N$ at each rebalancing, overweights small companies relative to their size and needs frequent, costly rebalancing as weights drift.
• A market-capitalization-weighted index weights each stock by its share of total market value, tracks aggregate investor wealth and rebalances automatically, but gives more weight to stocks whose prices have risen; a float-adjusted version uses market cap × float %.
• A fundamental-weighted index weights constituents by fundamentals such as earnings, dividends, cash flow, sales or book value and has a value tilt.
• Rebalancing resets weights to target after price moves and matters mainly for fundamental-weighted and equal-weighted indexes; reconstitution changes the constituents, with the divisor adjusted on that date.

Module 4.1: Discounted Cash Flow Valuation

• Every instrument here is valued as the present value of its expected cash flows, each discounted with $PV=FV/(1+r{)}^t$.
• A bond sells at a premium when its coupon rate is above its YTM, at par when they are equal and at a discount when the coupon rate is below the YTM.
• A perpetuity is worth $PMT/r$ one period before its first payment, and preferred stock is valued the same way, $D_p/r_p$.
• In an amortizing loan each payment's interest is the rate times the opening balance, so the interest part falls and the principal part rises over the life of the loan.
• The Gordon growth model $V_0=D_1/(r_e-g_c)$ uses next period's dividend and requires constant growth below the required return.
• In a multistage DDM the terminal value $P_n=D_{n+1}/(r_e-g_c)$ is discounted $n$ periods, together with the dividend paid at $n$.

Module 4.2: Implied Returns and Cash Flow Additivity

• The implied return on a zero-coupon bond is $(FV/PV{)}^{1/N}-1$; a coupon bond's YTM is found with the calculator (CPT I/Y).
• A bond's price and its YTM move in opposite directions.
• Rearranging the Gordon model gives the required return $r_e=D_1/P_0+g_c$ and the implied growth rate $g_c=r_e-D_1/P_0$.
• Under cash flow additivity the PV of a stream equals the sum of the PVs of its pieces, so an uneven stream can be replicated with annuities and zero-coupon bonds.
• No arbitrage sets the implied forward rate $1y1y=(1+S_2{)}^2/(1+S_1)-1$ and the forward exchange rate $\text{Forward}/\text{Spot}=(1+r_{\text{price}}{)}^T/(1+r_{\text{base}}{)}^T$.
• In a one-period binomial model the hedge ratio is $h=(c_u-c_d)/(S_u-S_d)$, the hedged portfolio earns the risk-free rate, and $c_0=h{S}_0$ minus the PV of the hedged portfolio.

Module 5.1: Measures of Central Tendency

• The arithmetic mean weights every observation equally and is sensitive to outliers, while the median, the middle value of sorted data (the average of the two middle values with an even count), is a robust estimator.
• The mode is the most frequent value; a dataset can have one mode, two (bimodal), several or none, and for continuous data the modal interval is reported.
• A trimmed mean drops a set percentage of the extreme observations, split equally between the two ends, and averages the rest; a winsorized mean replaces the extremes with percentile values and averages all $n$ observations.
• When there are only a few outliers, a winsorized mean measures the center better than the arithmetic mean or a trimmed mean does.
• A quantile is a cut-off value with a given share of the observations at or below it, and the interquartile range $Q_3-Q_1$ contains the central 50% of the data.
• Stationary data have a mean and variance that do not change over time; trending price levels are typically nonstationary, while their periodic percentage changes are much more likely to be stationary.

Module 5.2: Dispersion, Skewness, and Kurtosis

• The range, maximum minus minimum, uses only the two extreme observations, while the mean absolute deviation $\sum |X_i-\bar{X}|/n$ averages the unsigned deviations from the mean.
• Sample variance divides the sum of squared deviations from the mean by $n-1$ (Bessel's correction), which makes $s^2$ an unbiased estimator of ${\sigma }^2$; population variance divides by $N$.
• Taking the positive square root of the variance gives the standard deviation, which is in the same units as the data, while the variance is in squared units.
• Under the root of time law, with independent returns the variance over $T$ periods is $T{\sigma }^2$ and the standard deviation is $\sigma \sqrt{T}$, although long-horizon variance is often lower in practice.
• A positively skewed distribution has a long right tail with mode < median < mean, a negatively skewed one has a long left tail with mean < median < mode, and a symmetrical one has all three equal.
• A normal distribution has kurtosis 3; excess kurtosis is kurtosis − 3, a leptokurtic distribution (above 3) has fatter tails, and more negative skew and greater excess kurtosis both mean greater risk.

Module 5.3: Covariance, Correlation, and Alternative Measures of Dispersion

• Sample covariance is $\sum (X_i-\bar{X})(Y_i-\bar{Y})/(n-1)$; its sign shows the direction of co-movement, but its size depends on the units and it has no upper or lower bound.
• Correlation ${\rho }_{XY}={\text{Cov}}_{XY}/(s_X{s}_Y)$ has no units, lies between −1 and +1, and $\rho =0$ means no linear relationship.
• Correlation is not causation, outliers can create or destroy a correlation, and spurious correlation arises by chance or because both variables are related to a third variable.
• A scatter plot can reveal a nonlinear relationship that a correlation coefficient would miss.
• Semivariance uses only the outcomes below the mean, and the target downside deviation uses only the outcomes below a target $B$ but keeps the full sample size minus one in the denominator.
• The coefficient of variation $CV=s/\bar{X}$ measures risk per unit of mean return, is unit-free, and a lower CV is better.

Module 6.1: Probability Distributions and Expected Values

• A discrete random variable has a probability mass function with positive probabilities at individual outcomes; a continuous variable has a density, zero probability at any single point and probabilities assigned to intervals.
• A probability function is valid only if every $p(x)$ lies between 0 and 1 and the probabilities sum to 1.
• The CDF is $F(x)=P(X\le x)$, and $P(a<X\le b)=F(b)-F(a)$.
• Expected value is $E(X)=\sum P(x_i)x_i$; variance is $\sum P(x_i)[x_i-E(X){]}^2$ in squared units, and the standard deviation is its square root.
• Covariance shows only the direction of co-movement; correlation $\rho =\text{Cov}(X,Y)/({\sigma }_X{\sigma }_Y)$ is unit-free and lies between −1 and +1.

Module 6.2: Discrete and Continuous Probability Distributions

• A binomial variable counts successes in $n$ independent trials with a constant $p$: $E(X)=np$ and ${\sigma }^2=np(1-p)$; a single trial is a Bernoulli distribution.
• A Poisson variable counts events in an interval at a constant rate $\lambda$, and its mean and variance both equal $\lambda$.
• For a continuous uniform variable on $[a,b]$, $P(x_1\le X\le x_2)=(x_2-x_1)/(b-a)$.
• The normal distribution is symmetric, fully described by its mean and variance, and unbounded in both directions.
• If $\ln Y$ is normal, $Y$ is lognormal: bounded below by zero and positively skewed, which suits asset prices.
• The logistic distribution is symmetric like the normal but has fatter tails (excess kurtosis +1.2).

Module 6.3: Conditional Expectations and Bayesian Updating

• A joint probability is $P(A\text{ and }B)$, marginal probabilities are the row and column totals, and a conditional probability uses only the outcomes consistent with the condition.
• The conditional expectation is $E(X\mid S)=\sum P(x_i\mid S)x_i$, and the conditional variance is measured around $E(X\mid S)$.
• In a probability tree a path probability is the product of its branch probabilities.
• Bayes' formula is $P(E\mid I)=P(I\mid E)P(E)/P(I)$, where $P(I)$ adds the joint probabilities of the information with every state.
• $P(I\mid E)$ and $P(E\mid I)$ are generally different, and the posterior still depends on the prior.

Module 7.1: The Central Limit Theorem, Confidence Intervals, and Sampling

• By the central limit theorem, the mean of samples of size $n\ge 30$ from any population with a finite variance is approximately normal with mean $\mu$ and variance ${\sigma }^2/n$.
• The sample mean's standard error is $\sigma /\sqrt{n}$, or $s/\sqrt{n}$ when $\sigma$ is unknown.
• A confidence interval is the point estimate ± reliability factor × standard error, with a t reliability factor and $n-1$ df when $\sigma$ is unknown.
• A desirable estimator is unbiased, efficient and consistent.
• Parametric VaR is $z_{\alpha }\times \sigma \times$ portfolio value with the whole $\alpha$ in one tail; it is the minimum loss at the stated probability, not the worst possible loss.
• Simple random, systematic, stratified and cluster sampling are probability methods; convenience and judgmental sampling are nonprobability methods.

Module 7.2: Hypothesis Tests of the Population Mean

• $H_0$ always contains the equality and is usually the claim the researcher hopes to reject; $H_a$ is the claim the researcher seeks evidence for.
• The test statistic is (sample mean − ${\mu }_0$) ÷ standard error, following a t-distribution with $n-1$ df when $s$ is used.
• $H_0$ is rejected when the statistic falls beyond the critical value; otherwise the decision is to fail to reject, and the null is never accepted.
• A Type I error rejects a true null with probability $\alpha$; a Type II error fails to reject a false null with probability $\beta$; power is $1-\beta$.
• With the sample size unchanged, a lower $\alpha$ makes a Type II error more likely and reduces power.

Module 7.3: Other Parametric Hypothesis Tests

• Independent samples with equal variances use the pooled difference-in-means t-test with $n_1+n_2-2$ df; the same subjects measured twice use the paired comparisons test with $n-1$ df.
• A single variance is tested with ${\chi }^2=(n-1)s^2/{\sigma }_0^2$ and $n-1$ df; the chi-square distribution is right-skewed and bounded below by zero.
• Two variances are tested with $F=s_1^2/s_2^2$ and $n_1-1$, $n_2-1$ df.
• A correlation is tested with $t=r\sqrt{n-2}/\sqrt{1-r^2}$ and $n-2$ df; the statistic rises with $|r|$ and with $n$.
• The p-value is the smallest significance level at which $H_0$ can be rejected.
• p-hacking raises the frequency of Type I errors.

Module 7.4: Nonparametric Hypothesis Tests

• A nonparametric test is used for ranks or categories, when the assumptions of a parametric test do not hold, or when the question is not about a parameter.
• The Spearman rank correlation is $r_s=1-6\sum d_i^2/[n(n^2-1)]$, tested in large samples ($n>30$) with the same t-statistic as a Pearson correlation.
• In a chi-square test of independence the expected count is row total × column total ÷ total, ${\chi }^2=\sum (O-E{)}^2/E$ with $(r-1)(c-1)$ df, and only the right tail is used.
• A standardized residual is $(O-E)/\sqrt{E}$, and cells beyond ±2 are flagged by the rule of thumb.

Module 8.1: Portfolio Expected Return

• A portfolio's expected return is the weighted average $E(R_P)=\sum w_{i}E(R_i)$, with each weight equal to the asset's share of total portfolio value and the weights summing to 1.
• A portfolio's historical return for one period is $\sum w_{i,t-1}{R}_{i,t}$, using the weights at the start of the period.
• The cumulative (compound) return $\prod (1+R_{P,t})-1$ and the geometric average include compounding, while the sum and the arithmetic average of the period returns ignore it.
• Without rebalancing, market moves shift the weights (portfolio drift), so each period's return must use the drifted weights from start-of-period market values.
• Style drift is a portfolio moving away from its stated strategy, either through portfolio drift or through changes in the holdings themselves, such as a small-cap stock growing into the mid-cap category.
• Exam convention: drift raises the weights of high-return securities in an equal-weighted portfolio and enlarges already large allocations in a portfolio that is not equally weighted, so it behaves like a momentum strategy with possibly higher concentration risk; current practice: in any unrebalanced portfolio the outperformers gain weight, and whether concentration or risk rises depends on which assets outperform.

Module 8.2: Portfolio Risk Measures

• Two-asset portfolio variance is ${\sigma }_P^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\text{Cov}}_{1,2}$, and the portfolio standard deviation is its square root.
• Covariance can be written ${\text{Cov}}_{1,2}={\rho }_{1,2}{\sigma }_1{\sigma }_2$, which exam calculations use, although for forward-looking expected values the identity does not hold mechanically.
• With positive weights, a correlation of +1 gives a portfolio standard deviation equal to the weighted average of the standard deviations (no diversification benefit), and any lower correlation gives less, with a larger benefit as $\rho$ falls.
• With $\rho =-1$, ${\sigma }_P=|w_1{\sigma }_1-w_2{\sigma }_2|$, which is zero only when $w_1{\sigma }_1=w_2{\sigma }_2$.
• A negative covariance makes the cross term negative, so the portfolio is much less risky than the weighted average of the standard deviations suggests.

Module 8.3: Correlation and Diversification Benefits

• The diversification benefit grows as correlation falls: two-asset combinations lie on a straight line at $\rho =+1$, bow to the left when correlation is below +1, and include one zero-risk mix at $\rho =-1$.
• With $N$ assets the variance-covariance matrix holds $N$ variances and $(N^2-N)/2$ unique covariances.
• Expected portfolio variance from scenarios is ${\sum }_m{p}_m[R_{P,m}-E(R_P){]}^2$, with scenario probabilities that sum to 1.
• An equally weighted portfolio of $N$ assets has variance ${\bar{\sigma }}^2/N+[(N-1)/N]\overline{\text{Cov}}$, which approaches the average covariance as $N$ grows and is approximated for a large portfolio by $\bar{\rho }{\bar{\sigma }}^2$.
• Unsystematic risk is removed by diversification and is not rewarded with higher expected return; systematic risk is not removed and is rewarded.
• Roughly 20 to 30 well-chosen assets capture most of the diversification benefit, and beyond about 30 assets each new holding adds close to none.

Module 8.4: The Minimum-Variance Portfolio and the Efficient Frontier

• The minimum-variance frontier holds the lowest-risk portfolio for each level of expected return, and the global minimum-variance portfolio is its point of lowest risk.
• The efficient frontier is the upper part of the minimum-variance frontier from the global minimum-variance portfolio upward; points to the left of the frontier are unattainable and attainable points below the efficient frontier are inefficient.
• For two risky assets, $w_A=({\sigma }_B^2-{\text{Cov}}_{A,B})/({\sigma }_A^2+{\sigma }_B^2-2{\text{Cov}}_{A,B})$ and $w_B=1-w_A$ give the minimum-variance portfolio.
• The optimal risky (tangency) portfolio is where a line from the risk-free rate touches the efficient frontier, the efficient portfolio with the highest Sharpe ratio $(E(R_P)-R_f)/{\sigma }_P$.
• Combining the risk-free asset with portfolio $P$ gives $E(R_C)=w_{P}E(R_P)+(1-w_P)R_f$ and ${\sigma }_C=w_P{\sigma }_P$, with $w_P=(r_{\text{target}}-R_f)/(E(R_P)-R_f)$ for a target return and a weight above 100% meaning borrowing at the risk-free rate.
• Return-constrained optimization minimizes risk for a given return, risk-constrained optimization maximizes expected return for a given risk, and risk-adjusted return optimization seeks the highest Sharpe ratio.

Module 8.5: Risk Aversion

• Optimal portfolio theory assumes rational, risk-averse investors who maximize utility, a single-period horizon, no taxes or transaction costs, and borrowing and lending at a constant risk-free rate.
• A risk-seeking investor has $A<0$ and downward-sloping indifference curves, a risk-neutral investor has $A=0$ and flat curves, and a risk-averse investor has $A>0$ and upward-sloping curves.
• Utility is $U=E(r)-\frac{1}{2}A{\sigma }^2$ with the variance in decimals, so a higher coefficient of risk aversion $A$ means a bigger penalty for risk.
• Risk tolerance, the inverse of risk aversion, can change with age, wealth and economic conditions.
• A risk-averse investor's indifference curves slope upward and are convex, curves higher and to the left give higher utility, and a more risk-averse investor has steeper curves.

Module 8.6: Capital Allocation Line and Capital Market Line

• Under its assumptions (a riskless risk-free asset, a risky portfolio with return and risk above the risk-free rate, only these two assets, and unlimited borrowing and lending at the risk-free rate), the CAL is $E(R_C)=R_f+[(E(R_P)-R_f)/{\sigma }_P]{\sigma }_C$, with intercept $R_f$ and slope equal to the risky portfolio's Sharpe ratio.
• Points on the CAL between the intercept and the risky portfolio involve lending at the risk-free rate, and points beyond the risky portfolio involve borrowing.
• An investor's optimal portfolio is where the CAL touches the highest attainable indifference curve; a more risk-averse investor chooses a lower-risk point on the same line.
• The CML is the CAL whose risky portfolio is the market portfolio, which holds all risky assets weighted by market value and is mean-variance efficient.
• Under the mutual fund theorem, every investor can obtain an efficient portfolio by combining the risk-free asset with the market portfolio.
• A CAL allows heterogeneous expectations, so each investor can have a different CAL, while the CML assumes homogeneous expectations and is the same for all investors.

Module 8.7: The Capital Asset Pricing Model

• Because unsystematic risk can be diversified away at no cost, the market rewards investors only for bearing systematic risk.
• The CAPM is a single-factor model, $E(R_i)=R_f+{\beta }_i[E(R_M)-R_f]$, that assumes homogeneous expectations, utility-maximizing investors holding efficient portfolios, price takers and markets in equilibrium.
• Beta is $\text{Cov}(R_i,R_M)/{\sigma }_M^2={\rho }_{i,M}{\sigma }_i/{\sigma }_M$, and the beta of the market portfolio is 1.
• Exam convention: a beta above 1 indicates an expected return above the market's and a beta below 1 one below it; current practice: this ordering holds only when the market risk premium is positive.
• The SML plots expected return against beta with intercept $R_f$ and slope equal to the market risk premium and applies to any asset, while the CML plots against total risk and applies only to efficient portfolios.
• Portfolio theory is limited by non-normal returns, unstable correlations, imperfectly efficient markets, sensitivity to estimation error and a future that may differ from the past, and it ignores transaction costs and taxes.

Module 9.1: Historical Simulation

• Every simulation sets parameters, generates scenarios, evaluates them and compiles the results; historical simulation, bootstrapping and Monte Carlo simulation differ mainly in where the scenarios come from.
• Historical simulation applies actual past changes in prices, returns or risk factors to the current portfolio positions and assumes that past data represent the future.
• Historical simulation is nonparametric and captures skewness, excess kurtosis, fat tails and volatility clustering, but it can only produce outcomes from the lookback period and is weak when there are regime changes or unprecedented events.
• Missing historical data are filled with a proxy, a substitute instrument or a weighted combination of instruments with data available.
• VaR is the minimum loss over a specified period in the worst 5% (or 1%) of outcomes; the 95% historical VaR is the fifth percentile of the simulated P&L distribution, and daily VaR usually assumes a mean return of zero.
• Because a bond's duration shrinks as it approaches maturity, historical simulation for bonds applies past yield changes for the same remaining maturity to today's cash flows instead of using the bond's own past prices.

Module 9.2: Bootstrap Resampling

• Bootstrapping is a nonparametric method that uses the sample at hand as a proxy for the population and builds new datasets by drawing from it at random with replacement, so each of $n$ observations has a $1/n$ chance on every draw.
• In bootstrapping the same observation can be drawn several times within one scenario and can appear in many samples or in none.
• Historical simulation replays the past dataset as if it were the population, while bootstrapping builds new combinations of past observations, which reduces the risk of overfitting to one historical sequence.
• Bootstrapping can give the distribution of a European call's payoff at maturity under historical returns, which is not by itself the option's current no-arbitrage price, and can estimate the volatility of target date funds.
• Bootstrapping needs minimal assumptions and works with small samples, but it carries any bias in the original sample into every resample, cannot generate outcomes outside the observed range and is not suited to autocorrelated time series.

Module 9.3: Monte Carlo Simulation

• Monte Carlo simulation draws scenarios from probability distributions specified by the analyst and runs thousands of trials to build a simulated frequency distribution, with more trials giving more precise estimates.
• Monte Carlo simulation is used to estimate portfolio performance and to value securities without a closed-form pricing formula, such as Asian-style options, mortgage-backed securities and convertible bonds.
• Exam convention: a Monte Carlo value for a European call should match the Black-Scholes-Merton value, getting closer as more trials are run; current practice: this holds only when the simulated prices are lognormal with constant volatility and a risk-neutral expected return, with the average payoff discounted at the risk-free rate.
• The Cholesky decomposition maps independent standard normal draws into correlated normal variables using an assumed correlation; it does not estimate correlations or covariances.
• Monte Carlo simulation is flexible, forward-looking and handles nonnormal distributions and correlations, but it is complex, model-dependent, computationally intensive and data-hungry, and it gives a statistical estimate without the cause-and-effect insight of analytic methods.
• Historical simulation assumes past data represent the future, bootstrapping assumes the sample represents the population, and Monte Carlo simulation assumes the process generating outcomes can be modeled statistically.

Module 10.1: Linear Regression Basics

• Simple linear regression explains the variation of a dependent variable Y, $\sum (Y_i-\bar{Y}{)}^2$, with the variation of a single independent variable X.
• Ordinary least squares chooses the line that minimizes the sum of squared errors, $\text{SSE}=\sum (Y_i-{\hat{Y}}_i{)}^2$.
• The OLS estimates are ${\hat{b}}_1=\text{Cov}(X,Y)/{\sigma }_X^2$ and ${\hat{b}}_0=\bar{Y}-{\hat{b}}_1\bar{X}$, so the fitted line passes through the point of means.
• The slope is the expected change in Y for a one-unit rise in X, and the intercept is the predicted value of Y when X equals zero.
• The residual $Y_i-{\hat{Y}}_i$ is the gap between the actual and the fitted value; exam convention treats the residual, error term and disturbance term as the same thing, while current practice treats the residual as the sample estimate of the unobservable error term.
• Time series data follow one variable over time, cross-sectional data cover many subjects at one point in time, and panel data repeat cross-sectional observations over time.

Module 10.2: Analysis of Variance (ANOVA) and Goodness of Fit

• Simple linear regression assumes a linear relationship, homoskedastic residuals, residuals uncorrelated with one another, and normally distributed residuals.
• Residual variance that widens or narrows with X or over time is heteroskedasticity, and residuals correlated with one another, such as a recurring seasonal pattern, show autocorrelation.
• ANOVA splits total variation into explained and unexplained parts, $\text{SST}=\text{SSR}+\text{SSE}$.
• In a simple regression $\text{MSE}=\text{SSE}/(n-2)$, $\text{SEE}=\sqrt{\text{MSE}}$, and $R^2=\text{SSR}/\text{SST}=1-\text{SSE}/\text{SST}$, which equals $r^2$.
• $F=\text{MSR}/\text{MSE}$ with 1 and $n-2$ degrees of freedom is a one-tailed test, and with one independent variable it tests $H_0:b_1=0$, the same hypothesis as the slope's t-test.
• Regressing Y on an indicator variable gives a slope equal to the difference in the mean of Y between periods with and without the condition, and an intercept equal to the mean when the indicator is 0.

Module 10.3: Predicted Values and Functional Forms of Regression

• A predicted value is $\hat{Y}={\hat{b}}_0+{\hat{b}}_1{X}_p$, using the intercept and the sign of each term.
• The prediction interval for Y is $\hat{Y}\pm t_c{s}_f$ with $n-2$ degrees of freedom, where $s_f$ is the standard error of the forecast.
• The forecast is more precise when the SEE is smaller, the sample is larger, $X_p$ is closer to $\bar{X}$ and the independent variable has a greater variance.
• A log-lin model ($\ln Y$ on X) has a slope giving the relative change in Y for an absolute change in X, a lin-log model the absolute change in Y for a relative change in X, and a log-log model an elasticity.
• Exam convention: a higher $R^2$, a higher F-statistic and a lower SEE identify the better-fitting functional form; current practice: $R^2$ and SEE are directly comparable only when the dependent variable is the same.

Module 10.4: Estimating CAPM Values With Simple Linear Regression

• The CAPM is estimated by regressing the asset's excess return $R_i-R_f$ on the market risk premium $R_M-R_f$; the slope is the asset's beta and the intercept is zero if the CAPM holds exactly.
• A security market index proxies the market return, short-term government debt yields proxy the risk-free rate, and at least 30 observations is the usual minimum.
• Beta is tested with $t=({\hat{b}}_1-\text{hypothesized value})/s_{{\hat{b}}_1}$, and its confidence interval is the slope $\pm$ critical value $\times$ the slope's standard error.
• The expected return is $E(R_i)=R_f+\hat{\beta }[E(R_M)-R_f]$, and a change in inputs gives $\Delta E(R_i)=\Delta R_f+\hat{\beta }\Delta [E(R_M)-R_f]$.
• The $R^2$ of the CAPM regression is the share of the asset's excess-return variation that is systematic, and $1-R^2$ is the unsystematic share.
• Longer samples give more precise estimates but may include periods with a different beta, and more frequent data can be noisy for thinly traded assets.

• Financial data science applies quantitative and qualitative analysis to gain insight into a specific financial question, with main uses in evaluating investment opportunities, optimizing portfolios and mitigating risk.
• Big data can be structured (rows and columns), semi-structured (tagged, such as HTML code) or unstructured (social media posts, voice recordings, pictures, sensor output), and comes from traditional sources or from alternative sources: individuals, business processes and sensors.
• The characteristics of big data are volume, velocity (low latency means real time), variety of structures and, for financial data especially, veracity, meaning whether the data are reliable and credible.
• Data science processes data through capture, curation (ensuring data quality), storage, search and transfer, and a data lake keeps structured, semi-structured and unstructured data in their raw format.
• Data mining searches large datasets for patterns to predict outputs, while machine learning, a subcategory of AI, learns from input data without assumptions about their distribution and without human help.
• Supervised learning uses labeled inputs and outputs, unsupervised learning finds structure in unlabeled inputs, and reinforcement learning learns by trial and error from rewards and penalties.
• Overfitting comes from a model that is too complex and treats noise as a true pattern; underfitting comes from one that is not complex enough and treats true parameters as noise.
• The training dataset is the largest, about 60%–80% of the data, with validation and test sets splitting the rest equally; observations are normally assigned at random, but time series data keep their chronological order through a time-based split or rolling-window validation.

Module 12.1: Breakeven, Shutdown, and Scale

• In the short run at least one input is fixed; in the long run every input is variable.
• A price taker breaks even at a price equal to minimum ATC and shuts down in the short run when price falls below minimum AVC.
• When price is between AVC and ATC the firm keeps operating in the short run, because its loss is smaller than its fixed cost, and exits in the long run unless price is expected to rise.
• In any market structure: $TR\ge TC$ means stay; $TVC\le TR<TC$ means operate in the short run and exit in the long run if the shortfall is expected to persist; $TR<TVC$ means shut down.
• Marginal cost crosses AVC and ATC at their minimum points.
• The minimum efficient scale is the lowest point of LRATC; economies of scale make LRATC fall and diseconomies of scale make it rise.

Module 12.2: Characteristics of Market Structures

• Every firm maximizes profit where MR = MC; a price taker has $P=MR$, while a price searcher has $MR<P$.
• Monopolistic competition has many firms, differentiated products and low barriers, and earns zero economic profit in the long run with excess capacity.
• Oligopoly is defined by interdependence among a few large firms protected by high barriers to entry.
• Cournot firms decide simultaneously and the Stackelberg leader decides first; exam convention: they choose prices, the Cournot firm assuming its rival keeps last period's price and the Stackelberg leader charging a higher price and earning the larger share of total profit; current practice: they choose quantities and sell at one market price.
• In a Nash equilibrium no firm can gain by changing its choice alone, and the outcome need not maximize joint profit.
• The oligopoly price lies between the collusive (monopoly) price and the perfectly competitive price.

Module 12.3: Identifying Market Structures

• A market structure is identified by the number of firms, the barriers to entry, the nature of substitutes and the nature of competition.
• The N-firm concentration ratio is the sum of the N largest market shares; the HHI is the sum of their squared shares.
• The HHI is more sensitive than the concentration ratio to mergers among the largest firms.
• Neither measure accounts for barriers to entry or potential competition, and neither measures the elasticity of firm demand directly.

• The business cycle has four phases: trough (real GDP stops falling and growth turns positive), expansion (real GDP rising), peak (real GDP at its high for the cycle, then starting to fall) and contraction or recession (real GDP falling).
• The classical cycle plots the level of real GDP and the growth cycle its percentage gap from trend; the growth rate cycle turns earlier at peaks and troughs and, under the exam convention, is the preferred measure that shows GDP relative to a trend rate, while in current practice it tracks the growth rate itself, which can be compared with trend growth.
• A common rule of thumb dates a contraction from two consecutive quarters of falling real GDP and an expansion from two consecutive quarters of rising real GDP, while official dating bodies also weigh unemployment, industrial production and inflation.
• Credit cycles, swings in loan availability and interest rates, tend to amplify business cycles, can feed asset price bubbles and have on average lasted longer than business cycles.
• The inventory-sales ratio rises above normal late in an expansion as sales growth slows, leading firms to cut production, and falls below normal near the trough as sales pick up, leading firms to raise output.
• Firms first change how intensively they use existing workers and equipment, and only once the trend looks durable do they hire or lay off workers, invest in new capacity, or defer maintenance and delay replacing equipment.
• Durable goods spending is the most cyclical, services spending moderately cyclical and nondurable goods spending the least; imports rise with domestic GDP growth, while exports depend mainly on trading partners' growth.
• Leading indicators turn before peaks and troughs, coincident indicators turn at about the same time, and lagging indicators turn after a new phase is under way and confirm it.

Module 14.1: Fiscal Policy Objectives

• Fiscal policy is the government's use of taxes and government spending to steer economic activity, and monetary policy is the central bank's management of the quantity of money and credit.
• Both policies aim at price stability and economic growth, and only fiscal policy is a direct tool for redistributing income and wealth.
• Expansionary fiscal policy means higher spending or lower taxes, giving a larger deficit or smaller surplus; expansionary monetary policy is also called accommodative or easy, and contractionary monetary policy restrictive or tight.
• Automatic stabilizers such as unemployment compensation and the progressive income tax push the budget toward deficit in a recession and toward surplus in a boom without new legislation, while discretionary fiscal policy needs legislation and is slowed by policy lags.
• Exam convention: with tax rates held constant, the debt ratio rises when the real interest rate on government debt exceeds real GDP growth and falls when it is below; current practice: this holds when the primary budget is balanced, since a primary deficit or surplus can reverse the result.
• Concerns about deficits are higher future taxes, a loss of market confidence and crowding out of private investment; arguments against concern include debt held by citizens, debt that finances productive investment, needed tax reform, Ricardian equivalence and an economy below full capacity.

Module 14.2: Fiscal Policy Tools and Implementation

• Fiscal spending tools are transfer payments (not counted in GDP), current spending and capital spending, and revenue tools are direct taxes on income or wealth and indirect taxes on goods and services; indirect taxes can be changed quickly, while direct taxes, transfers and capital spending take time.
• Government spending raises aggregate demand more than an equal tax cut, because households save part of a tax cut, and tax cuts aimed at low-income households, who have a higher MPC, work better.
• The fiscal multiplier is $1/[1-MPC(1-t)]$, which rises with the MPC and falls with the tax rate.
• Exam convention: equal increases in government spending and taxes leave the budget balanced and raise aggregate demand, so the balanced budget multiplier is positive; current practice: with an income tax the higher output also raises tax receipts, but the direction of the output effect is unchanged.
• Recognition, action and impact lags separate the need for discretionary fiscal policy from its effect, so a badly timed policy can destabilize the economy.
• The stance is judged from the change in the structural (cyclically adjusted) deficit, the deficit current policies would produce at full employment: a rise means expansionary policy, while a change in the actual deficit that leaves it unchanged reflects the cycle.

Module 15.1: Central Bank Objectives and Tools

• A central bank supplies currency, acts as banker to the government and other banks, regulates and supervises the payments system, is lender of last resort, holds gold and foreign exchange reserves and conducts monetary policy; under the exam convention it is the sole supplier of money, while in current practice its monopoly covers currency and bank reserves and commercial banks create most deposit money.
• Controlling inflation to promote price stability is a central bank's primary objective, and some central banks also aim for stable exchange rates, full employment, sustainable growth and moderate long-term interest rates.
• Most developed-country central banks aim for inflation in a range around 2% to 3%, not zero, because swings around a zero target would often mean deflation.
• Exam convention: open market operations, the policy rate and reserve requirements are the three working tools, used expansionarily by cutting the rate or requirements and buying securities; current practice: the Federal Reserve has set reserve requirements to zero and steers the federal funds rate mainly through the rate it pays on reserve balances.
• The federal funds rate is the market rate US banks charge each other for overnight loans of reserves, which the Federal Reserve targets, while the discount rate is what a bank pays to borrow from the Federal Reserve.
• A policy-rate change reaches the price level through other short-term rates, asset prices, expectations and exchange rates; a rate increase lowers aggregate demand and appreciates the currency, and if money neutrality holds, monetary policy has no long-run effect on real output.

Module 15.2: Monetary Policy Effects and Limitations

• An effective central bank has independence, credibility and transparency; operational independence means it sets the policy rate alone, and target independence means it also defines the inflation measure, the target and the horizon for reaching it.
• Inflation targeting, the most widely used regime, keeps expected inflation inside a band, most commonly 2% ± 1%, while an exchange rate target makes the central bank buy its currency with foreign reserves when it falls below target and sell it when it rises above, so the pegging country ends up with the anchor country's inflation rate.
• The neutral interest rate equals the real trend rate of economic growth plus the inflation target, and a policy rate above it is contractionary and below it expansionary; the exam convention describes the neutral rate as the money supply growth rate that neither speeds up nor slows growth, while current practice treats it as the policy rate that neither stimulates nor restrains the economy.
• Monetary policy can fail when long-term rates move against short-term rates because of expected inflation, in a liquidity trap where demand for money is very elastic, and when banks will not lend despite excess reserves, which led central banks to quantitative easing.
• Exam convention: the nominal policy rate cannot be cut below zero, leaving little room to stimulate; current practice: some central banks have set slightly negative rates; either way deflation is harder to reverse than inflation.
• With expansionary fiscal and contractionary monetary policy, interest rates rise and the public sector's share of GDP rises, while contractionary fiscal and expansionary monetary policy lower rates and let the private sector grow.

• Geopolitics studies interactions among nations, involving state actors (national governments) and nonstate actors (companies, nongovernment organizations and individuals), and a country's cooperation varies along a spectrum, driven by its national interests in security, economic needs and culture.
• Globalization is the long-run integration of economic activity and cultures across the world, and nationalism here means a country putting its own economic interests first, alone or in competition with others.
• The IMF promotes international monetary cooperation, balanced growth of trade and exchange stability and makes resources available, with adequate safeguards, to members with balance of payments difficulties; the World Bank fights poverty with loans, credits and grants to developing countries; the WTO, the sole international body for the rules of trade between nations, runs a dispute settlement process.
• Geopolitical risk changes the risk premium on a country's or region's assets and is described by its likelihood, its impact and its velocity, meaning how quickly investment values reflect it.
• A black swan is a low-likelihood exogenous event with substantial short-term effects; long-horizon investors usually need not react to one and should instead analyze medium- and low-velocity risks.
• National security tools are classified as active or threatened, while economic and financial tools are classified as cooperative or noncooperative, and sanctions count as a financial tool.
• Investors should focus on geopolitical risks with potentially high impact, judge whether their effects are discrete or broad, estimate them with scenario analysis and track their likelihood with signposts.

• A country has a comparative advantage in a good when its opportunity cost of producing it is lower than other countries', and specializing according to comparative advantage raises total output.
• Newer trade models add gains from economies of scale, more variety for consumers, stronger competition and a better allocation of resources.
• Consumers of imported goods and export industries gain from trade while import-competing producers and workers lose, but overall gains exceed losses, especially in the long run, and both trading partners share the net gains.
• Infant industry and national security arguments for trade restrictions have some support among economists, while protecting domestic jobs or industries has little or none.
• Tariffs, import quotas and VERs reduce imports, raise the domestic price, domestic quantity supplied and producer surplus, lower consumer surplus and reduce national welfare, with the possible exception of a tariff or quota imposed by a large importing country.
• A quota whose licenses the government sells for full value has the same outcome as an equivalent tariff, while free licenses to foreign exporters send the quota rents abroad, and a VER causes the same welfare loss as an equivalent quota with free licenses to foreign exporters.
• An export subsidy raises the domestic price by the full subsidy in a small exporting country, while a large exporter's subsidy lowers the world price, benefiting foreign consumers and hurting foreign producers.
• A free trade area removes barriers among members, a customs union adds common restrictions on nonmembers, a common market adds free movement of labor and capital, an economic union adds common institutions and economic policy, and a monetary union adds a single currency.

Module 18.1: The Foreign Exchange Market

• The FX market serves trade flows and the much larger capital flows; a position that reduces an existing currency risk is hedging, and one that creates or increases currency risk is speculating.
• The sell side is the large multinational dealer banks, and the buy side is corporations, real money and leveraged accounts, governments and central banks, and the retail market; the exam convention separates real money from leveraged accounts by derivatives use, while current practice separates them by leverage.
• A P/B quote states how many units of the price currency buy one unit of the base currency, so a higher quote means the base currency has appreciated, and the quote is direct for an investor whose home currency is the price currency.
• A spot rate is for immediate delivery (usually two business days), and a forward rate is agreed today for an exchange of fixed amounts on a specified future date.
• The real P/B rate equals the nominal P/B rate × $CP{I}_{\text{base}}/CP{I}_{\text{price}}$, so it rises with the nominal rate and the base-country price level and falls with the price-country price level.
• The base currency changes by new P/B ÷ old P/B − 1 and the price currency by old P/B ÷ new P/B − 1, and the two percentages differ in size.

Module 18.2: Managing Exchange Rates

• The IMF lists two regimes for countries without their own currency (formal dollarization and monetary union) and seven for countries with one, and moving toward floating generally gives more room for independent monetary policy.
• A currency board explicitly commits to swap domestic currency for a named foreign currency at a fixed rate, issuing currency only when fully backed, while a conventional fixed peg allows ±1% margins and a target zone wider bands such as ±2%.
• A crawling peg adjusts the peg rate itself, crawling bands widen the band over time as a step toward floating, managed floating reacts to indicators without a target rate, and an independent float uses intervention only to slow the rate of change.
• A currency appreciation raises imports and cuts exports, but these trade effects build up slowly; capital flows dominate exchange rate moves in the short and intermediate term, and trade flows matter more in the long term.
• Under $(X-M)\equiv (S-I)+(T-G)$, a trade deficit is matched by a capital account surplus, and shrinking it on a lasting basis requires domestic spending to fall relative to income.
• Capital restrictions aim to reduce the volatility of domestic asset prices, maintain fixed exchange rates, keep domestic interest rates low and protect strategic industries from foreign ownership.