Quantitative Methods formula sheet.

Where $P_0$ = price at the start, $P_1$ = price at the end, $I_1$ = income (dividend, coupon) received during the period.

The total return over one holding period: price change plus income, as a share of what you paid. It is the building block for multi-period, money-weighted and time-weighted returns.

Exam tip Divide by the starting price, not the ending price, and do not annualise it unless the question asks: an HPR covers whatever period was held.

Lesson: Holding period return →All Rates and Returns formulas

Where $R_t$ = return in period t, $T$ = number of periods, $\prod$ = product of the growth factors.

The constant compound rate per period that turns the starting value into the actual ending value. Use it to describe past multi-period performance; the arithmetic mean is the better estimate of one future period's expected return.

Exam tip Convert each return to $1+R$ before multiplying. The geometric mean is never above the arithmetic mean, and the gap widens as returns become more volatile.

Lesson: Arithmetic and geometric mean returns →All Rates and Returns formulas

Where $CF_t$ = net cash flow at time t from the investor's view (deposits negative; withdrawals, income and ending value positive), $\text{IRR}$ = money-weighted return per period.

The IRR of the investor's own cash flows, so it rises or falls with the size and timing of deposits and withdrawals. It measures the investor's actual experience and suits cases where the investor, not the manager, controls the flows.

Exam tip BA II Plus: enter flows in the CF worksheet (deposits as negatives, ending value positive), then IRR CPT. If the periods are not years, annualise the periodic IRR.

Lesson: Money-weighted return →All Rates and Returns formulas

Where $R_i$ = holding period return of sub-period i, $k$ = number of sub-periods in the year, split at each external cash flow.

Links sub-period returns so that the size and timing of client cash flows have no effect. It is the preferred measure for evaluating a manager who does not control when money comes in or goes out.

Exam tip Start a new sub-period at every deposit or withdrawal. If the investor adds money just before a strong period, the MWR beats the TWR; for several years, take the geometric mean of the annual TWRs.

Lesson: Time-weighted vs money-weighted return →All Rates and Returns formulas

Where $R_{\text{period}}$ = return over the period, $c$ = number of such periods in a year (365/days, 52/weeks, 12/months).

Converts a return over any period into a compounded annual rate, so returns over different horizons can be compared. Typical items give a 90-day, weekly or 18-month return and ask for the annual equivalent.

Exam tip For periods longer than a year, $c$ is below 1: an 18-month return uses $c = 12/18$. Compound, do not simply multiply the periodic return by $c$.

Lesson: Annualised and continuously compounded returns →All Rates and Returns formulas

Where $P_t$, $P_{t+1}$ = prices at the start and end, $R_{t,t+1}$ = holding period return, $r_{t,t+1}$ = log return.

The log return. Log returns over consecutive periods simply add to give the log return for the whole horizon, which makes multi-period work easy. For any non-zero return it is below the matching holding period return.

Exam tip Go back to a holding period return with $R = e^{r} - 1$. Log returns add across periods; holding period returns compound and must never be added.

Lesson: Annualised and continuously compounded returns →All Rates and Returns formulas

Where $R_s$ = stated (quoted) annual rate, $m$ = compounding periods per year, $e$ = base of the natural log.

Turns a quoted annual rate into the rate actually earned over a year once compounding is counted. Use it to compare deposits or loans that compound at different frequencies: highest EAR is best for the lender, lowest for the borrower.

Exam tip More frequent compounding raises the EAR, with continuous compounding as the upper limit; EAR equals the stated rate only when $m = 1$. BA II Plus: ICONV worksheet.

Lesson: Present value and discount (zero-coupon) bonds →All Time Value of Money in Finance formulas

Where $PMT$ = coupon per period, $r$ = yield to maturity per period, $N$ = periods to maturity, $FV$ = principal repaid at maturity.

A bond's price is the present value of its coupons plus its principal, all discounted at one rate, the YTM. Solve for the price given the YTM, or for $r$ given the price. A price above par means the coupon rate is above the YTM.

Exam tip Keep everything per period: for semiannual coupons use half the annual coupon, half the YTM and twice the years. BA II Plus: N, I/Y, PMT, FV, then CPT PV.

Lesson: Coupon bonds and perpetual bonds →All Time Value of Money in Finance formulas

Where $D_0$ = dividend just paid, $D_1$ = next dividend, $r$ = required return, $g$ = constant dividend growth rate.

Values a share as a growing perpetuity of dividends. Rearranged, it gives the required return implied by the price (dividend yield plus g) or the implied growth rate. It also gives the terminal value in a two-stage model.

Exam tip The numerator is next period's dividend: grow $D_0$ by one period first. The model only works when $r > g$.

Lesson: Valuing shares with dividend discount models →All Time Value of Money in Finance formulas

Where $r_1$, $r_2$ = 1-year and 2-year spot rates, $F_{1,1}$ = 1-year rate starting in 1 year.

The no-arbitrage forward rate: investing for two years at the 2-year spot rate must give the same result as investing for one year and rolling over at the forward rate. Exam items ask for a forward rate from spot rates or the reverse.

Exam tip Compound, do not average: $2r_2 - r_1$ is only an approximation. For longer horizons, chain the growth factors the same way and take the root.

Lesson: Cash flow additivity, no arbitrage and forward rates →All Time Value of Money in Finance formulas

Where $X_i$ = observation i, $\bar X$ = sample mean, $n$ = number of observations.

The standard measure of total risk: the typical distance of returns from their mean, in the same units as the data. The exam asks you to compute it from a short return series or to compare the dispersion of two funds.

Exam tip Divide by $n - 1$ for a sample, $n$ for a population. BA II Plus: DATA, then STAT gives Sx (sample) and $\sigma$x (population).

Lesson: Range, mean absolute deviation, variance and standard deviation →All Statistical Measures of Asset Returns formulas

Where $B$ = target return, $X_i$ = observation i, $n$ = total number of observations.

Measures only the risk of falling short of a target return, ignoring variation above it. Use it when an investor cares about returns below a minimum, especially when returns are skewed.

Exam tip Deviations are measured from the target $B$, not from the mean. Only observations below $B$ enter the sum, but the denominator still uses the full sample: $n - 1$.

Lesson: Target downside deviation and coefficient of variation →All Statistical Measures of Asset Returns formulas

Where $s$ = sample standard deviation, $\bar X$ = sample mean.

Risk per unit of mean return. Because it is scale-free, it compares the relative dispersion of assets with very different mean returns. A lower CV means less risk for each unit of return.

Exam tip Lower is better for the CV, the opposite of the Sharpe ratio. The CV is not meaningful when the mean is negative or close to zero.

Lesson: Target downside deviation and coefficient of variation →All Statistical Measures of Asset Returns formulas

Where $E(X \mid S_i)$ = expected value of X in scenario $S_i$, $P(S_i)$ = probability of the scenario; scenarios are mutually exclusive and exhaustive.

The unconditional expected value is the probability-weighted average of the scenario expected values. This is how you roll a probability tree back to one number, such as an expected EPS across economic scenarios.

Exam tip Check that the scenario probabilities sum to 1. On a tree, multiply probabilities along each path, then weight each end value by its path probability.

Lesson: Conditional expected values and variances →All Probability Trees and Conditional Expectations formulas

Where $P(E)$ = prior probability of the event, $P(I \mid E)$ = probability of the information given the event, $P(I)$ = unconditional probability of the information.

Updates a prior probability once new information arrives, giving the posterior. Typical items: the probability that a firm beats earnings given an analyst upgrade, or that a borrower defaults given a warning signal.

Exam tip Find $P(I)$ first with the total probability rule: $P(I) = P(I \mid E)P(E) + P(I \mid E^C)P(E^C)$. A tree or a table of joint probabilities avoids mix-ups.

Lesson: Bayes' formula: updating a probability with new information →All Probability Trees and Conditional Expectations formulas

Where $w_1$, $w_2$ = portfolio weights, $\sigma_1$, $\sigma_2$ = standard deviations, $\rho_{12}$ = correlation between the two returns.

Portfolio risk depends on the weights, each asset's risk and how the two move together. With any correlation below +1, the portfolio standard deviation is below the weighted average of the two: that is the benefit of diversification.

Exam tip Take the square root to get $\sigma_p$. If a covariance is given, use $2w_1w_2\text{Cov}_{12}$ directly. The expected return is just the weighted average; correlation does not affect it.

Lesson: Correlation and diversification →All Portfolio Mathematics formulas

Where $E(R_p)$ = expected portfolio return, $R_L$ = threshold (minimum acceptable) return, $\sigma_p$ = portfolio standard deviation.

Choose the portfolio with the highest ratio: with normal returns, it has the lowest probability of a return below the threshold. When the threshold is the risk-free rate, the ratio equals the Sharpe ratio.

Exam tip Shortfall probability $= N(-\text{SFRatio})$. If the client must withdraw cash without touching principal, $R_L$ = withdrawal divided by current portfolio value.

Lesson: Shortfall risk and Roy's safety-first criterion →All Portfolio Mathematics formulas

Where $\bar X$ = sample mean, $s$ = sample standard deviation, $n$ = sample size, $t_{\alpha/2,\,n-1}$ = critical t with n − 1 degrees of freedom.

A range built so that, across repeated samples, $1 - \alpha$ of such intervals contain the population mean. Use t when the population variance is unknown (the usual case) and z when $\sigma$ is known.

Exam tip $s/\sqrt{n}$ is the standard error. A larger sample narrows the interval; a higher confidence level widens it. z values: 1.645 (90%), 1.96 (95%), 2.58 (99%).

Lesson: Confidence intervals for the mean →All Estimation and Inference formulas

Where $\bar X$ = sample mean, $\mu_0$ = hypothesised mean, $s$ = sample standard deviation, $n$ = sample size.

How many standard errors the sample mean lies from the hypothesised value. Reject $H_0$ when the statistic is beyond the critical value, or equivalently when the p-value is below the significance level.

Exam tip Use n − 1 degrees of freedom. For a one-tailed test put all of $\alpha$ in the tail named by $H_a$; for a two-tailed test put $\alpha/2$ in each tail.

Lesson: Testing a single mean →All Hypothesis Testing formulas

Where $s^2$ = sample variance, $\sigma_0^2$ = hypothesised variance, $n$ = sample size.

Tests whether a single population variance (for example a fund's risk) equals, exceeds or falls below a stated value. It assumes a normal population; the chi-square distribution is not symmetric and cannot be negative.

Exam tip Square any standard deviations first. In a two-tailed test the lower and upper critical values are not mirror images, so look up each one separately.

Lesson: Testing variances: chi-square and F →All Hypothesis Testing formulas

Where $s_1^2$, $s_2^2$ = sample variances of two independent samples, $n_1$, $n_2$ = sample sizes.

Tests whether two normal populations have equal variances, for example whether a fund's volatility changed between two periods or differs between two managers.

Exam tip Put the larger variance in the numerator so F is at least 1 and only the upper critical value matters. The numerator's degrees of freedom are listed first.

Lesson: Testing variances: chi-square and F →All Hypothesis Testing formulas

Where $r$ = sample correlation, $n$ = number of paired observations.

Tests whether the population correlation is zero. The statistic grows with both the size of r and the sample size, so a weak correlation can be significant in a large sample.

Exam tip Degrees of freedom are n − 2, not n − 1. Reject $H_0: \rho = 0$ when |t| is above the critical value.

Lesson: The t-test for a Pearson correlation →All Parametric and Non-Parametric Tests of Independence formulas

Where $d_i$ = rank of X minus rank of Y for pair i, $n$ = number of pairs.

A non-parametric correlation computed on ranks. Use it when the data are not normal, contain outliers or are only ordinal. Like Pearson's r, it lies between −1 and +1.

Exam tip Rank each variable separately before taking differences. For n above 30, test it with the same t-formula as a Pearson correlation, with n − 2 degrees of freedom.

Lesson: The Spearman rank correlation test →All Parametric and Non-Parametric Tests of Independence formulas

Where $O_{ij}$ = observed count in a cell, $E_{ij}$ = expected count under independence, $m$ = number of cells, $r$, $c$ = numbers of row and column categories.

Tests whether two categorical classifications, such as fund style and fund size, are independent, by comparing observed counts with the counts expected if they were. A large statistic is evidence against independence.

Exam tip Expected count = row total × column total ÷ overall total. The test is always right-tailed, with (rows − 1) × (columns − 1) degrees of freedom.

Lesson: Contingency tables and the chi-square test of independence →All Parametric and Non-Parametric Tests of Independence formulas

Where $X$ = independent variable, $Y$ = dependent variable, $\bar X$, $\bar Y$ = their sample means.

The least squares slope: the expected change in Y for a one-unit change in X. The intercept follows as $\hat b_0 = \bar Y - \hat b_1\bar X$, so the fitted line always passes through the two means.

Exam tip Divide by the variance of X, the independent variable, not of Y. The slope always has the same sign as the correlation.

Lesson: The regression model and least squares →All Simple Linear Regression formulas

Where $\hat b_1$ = estimated slope, $B_1$ = hypothesised slope, $s_{\hat b_1}$ = standard error of the slope, $n$ = number of observations.

Tests whether the slope differs from a stated value, most often zero (does X help explain Y?), but also values such as 1 when testing whether a stock's beta equals the market's.

Exam tip Regression output prints t for $B_1 = 0$ only; for any other hypothesised value, recompute it yourself. Degrees of freedom are n − 2.

Lesson: Testing the slope: t-test, correlation test and F-test →All Simple Linear Regression formulas

Where $SST$ = total variation of Y, $SSR$ = explained, $SSE$ = unexplained, $MSR = SSR/1$, $MSE = SSE/(n-2)$, $t_{b_1}$ = slope t-statistic.

The fit measures read from an ANOVA table. $R^2$ is the share of the variation in Y explained by X; the SEE is the typical size of a residual, so lower means a tighter fit; F tests whether the slope is zero.

Exam tip SST = SSR + SSE. With one regressor, $R^2$ equals the squared correlation and F equals the slope's t squared, with (1, n − 2) degrees of freedom.

Lesson: Integrated cases: reading a full regression printout →All Simple Linear Regression formulas

Where $SEE$ = standard error of the estimate, $X_f$ = value of X used for the forecast, $\bar X$ = mean of X, $n$ = number of observations.

Used to build the prediction interval $\hat Y_f \pm t_{\alpha/2}\,s_f$ around a forecast of Y. It is always larger than the SEE, because it adds the uncertainty in the estimated intercept and slope.

Exam tip The interval widens as $X_f$ moves away from the mean of X. Use the critical t with n − 2 degrees of freedom.

Lesson: Integrated cases: reading a full regression printout →All Simple Linear Regression formulas