The real risk-free rate is a theoretical rate on a single-period loan when there is no expectation of inflation. Nominal risk-free rate = real risk-free rate + expected inflation rate.

Securities may have several risks, and each increases the required rate of return. These include default risk, liquidity risk, and maturity risk.

The required rate of return on a security = real risk-free rate + expected inflation + default risk premium + liquidity premium + maturity risk premium.

Future value: FV = PV(1 + I/Y)^N

Present value: PV = FV / (1 + I/Y)^N

An annuity is a series of equal cash flows that occurs at evenly spaced intervals over time. Ordinary annuity cash flows occur at the end of each time period. Annuity due cash flows occur at the beginning of each time period.

Perpetuities are annuities with infinite lives (perpetual annuities):

[image]

The present (future) value of any series of cash flows is equal to the sum of the present (future) values of the individual cash flows.

The NPV is the present value of a project’s future cash flows, discounted at the firm’s cost of capital, less the project’s cost. IRR is the discount rate that makes the NPV = 0 (equates the PV of the expected future cash flows to the project’s initial cost).

The NPV rule is to accept a project if NPV > 0; the IRR rule is to accept a project if IRR > required rate of return. For an independent (single) project, these rules produce the exact same decision.

For mutually exclusive projects, IRR rankings and NPV rankings may differ due to differences in project size or in the timing of the cash flows. Choose the project with the higher NPV as long as it is positive.

A project may have multiple IRRs or no IRR.

The holding period return (or yield) is the total return for holding an investment over a certain period of time and can be calculated as:

[image]

The money-weighted rate of return is the IRR calculated using periodic cash flows into and out of an account and is the discount rate that makes the PV of cash inflows equal to the PV of cash outflows.

The time-weighted rate of return measures compound growth. It is the rate at which $1 compounds over a specified performance horizon.

If funds are added to a portfolio just before a period of poor performance, the money-weighted return will be lower than the time-weighted return. If funds are added just prior to a period of high returns, the money-weighted return will be higher than the time-weighted return.

The time-weighted return is the preferred measure of a manager’s ability to select investments. If the manager controls the money flows into and out of an account, the money-weighted return is the more appropriate performance measure.

Given a $1,000 T-bill with 100 days to maturity and a discount of $10 (price of $990):

• Bank discount yield = [image]
• Holding period yield = [image]
• Effective annual yield = [image]
• Money market yield = [image]

Given a money market security with n days to maturity:

• Holding period yield = [image]
• Money market yield = holding period yield × [image]
• Effective annual yield = (1 + holding period yield)^365/n− 1
• Holding period yield = (1 + effective annual yield)^n/365− 1
• Bond equivalent yield = [(1 + effective annual yield)^½ − 1] × 2

Descriptive statistics summarize the characteristics of a data set; inferential statistics are used to make probabilistic statements about a population based on a sample.

A population includes all members of a specified group, while a sample is a subset of the population used to draw inferences about the population.

Data may be measured using different scales.

• Nominal scale—data is put into categories that have no particular order.
• Ordinal scale—data is put into categories that can be ordered with respect to some characteristic.
• Interval scale—differences in data values are meaningful, but ratios, such as twice as much or twice as large, are not meaningful.
• Ratio scale—ratios of values, such as twice as much or half as large, are meaningful, and zero represents the complete absence of the characteristic being measured.

Any measurable characteristic of a population is called a parameter.

A characteristic of a sample is given by a sample statistic.

A frequency distribution groups observations into classes, or intervals. An interval is a range of values.

Relative frequency is the percentage of total observations falling within an interval.

Cumulative relative frequency for an interval is the sum of the relative frequencies for all values less than or equal to that interval’s maximum value.

The arithmetic mean is the average. [image]. Population mean and sample mean are examples of arithmetic means.

The geometric mean is used to find a compound growth rate. [image]

The weighted mean weights each value according to its influence. [image]

The harmonic mean can be used to find an average purchase price, such as dollars per share for equal periodic investments.

[image]

The median is the midpoint of a data set when the data is arranged from largest to smallest.

The mode of a data set is the value that occurs most frequently.

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies. Examples of quantiles include:

• Quartiles—the distribution is divided into quarters.
• Quintile—the distribution is divided into fifths.
• Decile—the distribution is divided into tenths.
• Percentile—the distribution is divided into hundredths (percents).

The range is the difference between the largest and smallest values in a data set.

Mean absolute deviation (MAD) is the average of the absolute values of the deviations from the arithmetic mean:

[image]

Variance is defined as the mean of the squared deviations from the arithmetic mean or from the expected value of a distribution.

• Population variance = [image]where μ = population mean and N = size.
• Sample variance = [image] where X = sample mean and n = sample size.

Standard deviation is the positive square root of the variance and is frequently used as a quantitative measure of risk.

Chebyshev’s inequality states that the proportion of the observations within k standard deviations of the mean is at least 1 − 1/k² for all k > 1. It states that for any distribution, at least:

36% of observations lie within +/− 1.25 standard deviations of the mean.
56% of observations lie within +/− 1.5 standard deviations of the mean.
75% of observations lie within +/− 2 standard deviations of the mean.
89% of observations lie within +/− 3 standard deviations of the mean.
94% of observations lie within +/− 4 standard deviations of the mean.

The coefficient of variation for sample data, CV = [image], is the ratio of the standard deviation of the sample to its mean (expected value of the underlying distribution).

The Sharpe ratio measures excess return per unit of risk:

[image]

Skewness describes the degree to which a distribution is not symmetric about its mean. A right-skewed distribution has positive skewness. A left-skewed distribution has negative skewness.

Sample skew with an absolute value greater than 0.5 is considered significantly different from zero.

For a positively skewed, unimodal distribution, the mean is greater than the median, which is greater than the mode.

For a negatively skewed, unimodal distribution, the mean is less than the median, which is less than the mode.

Kurtosis measures the peakedness of a distribution and the probability of extreme outcomes (thickness of tails).

• Excess kurtosis is measured relative to a normal distribution, which has a kurtosis of 3.
• Positive values of excess kurtosis indicate a distribution that is leptokurtic (fat tails, more peaked) so that the probability of extreme outcomes is greater than for a normal distribution.
• Negative values of excess kurtosis indicate a platykurtic distribution (thin tails, less peaked).
• Excess kurtosis with an absolute value greater than 1 is considered significant.

A random variable is an uncertain value determined by chance.

An outcome is the realization of a random variable.

An event is a set of one or more outcomes. Two events that cannot both occur are termed "mutually exclusive" and a set of events that includes all possible outcomes is an "exhaustive" set of events.

The two properties of probability are:

• The sum of the probabilities of all possible mutually exclusive events is 1.
• The probability of any event cannot be greater than 1 or less than 0.

A priori probability measures predetermined probabilities based on well-defined inputs; empirical probability measures probability from observations or experiments; and subjective probability is an informed guess.

Unconditional probability (marginal probability) is the probability of an event occurring.

Conditional probability, P(A | B), is the probability of an event A occurring given that event B has occurred.

The multiplication rule of probability is used to determine the joint probability of two events:

P(AB) = P(A | B) × P(B)

The addition rule of probability is used to determine the probability that at least one of two events will occur:

P(A or B) = P(A) + P(B) − P(AB)

The total probability rule is used to determine the unconditional probability of an event, given conditional probabilities:

P(A) = P(A | B₁)P(B₁) + P(A | B₂)P(B₂) +...+ P(A | B_N)P(B_N)

where B₁, B₂,...B_N is a mutually exclusive and exhaustive set of outcomes.

The joint probability of two events, P(AB), is the probability that they will both occur. P(AB) = P(A | B) × P(B). For independent events, P(A | B) = P(A), so that P(AB) = P(A) × P(B).

The probability that at least one of two events will occur is P(A or B) = P(A) + P(B) − P(AB). For mutually exclusive events, P(A or B) = P(A) + P(B), since P(AB) = 0.

The joint probability of any number of independent events is the product of their individual probabilities.

The probability of an independent event is unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occurrence of another event. Events A and B are independent if and only if:

P(A | B) = P(A), or equivalently, P(B | A) = P(B)

Using the total probability rule, the unconditional probability of A is the probability weighted sum of the conditional probabilities:

[image]

where B_i is a set of mutually exclusive and exhaustive events.

Conditional expected values depend on the outcome of some other event.

Forecasts of expected values for a stock’s return, earnings, and dividends can be refined, using conditional expected values, when new information arrives that affects the expected outcome.

A tree diagram shows the probabilities of two events and the conditional probabilities of two subsequent events.

[image]

Covariance measures the extent to which two random variables tend to be above and below their respective means for each joint realization. It can be calculated as:

[image]

Correlation is a standardized measure of association between two random variables; it ranges in value from –1 to +1 and is equal to [image]

The expected value of a random variable, E(X), equals [image]

The variance of a random variable, Var(X), equals [image]

Standard deviation: [image]

The expected returns and variance of a 2-asset portfolio are given by:

[image]

Given the joint probabilities for X_i and Y_i, i.e., P(X_iY_i), the covariance is calculated as:

[image]

Bayes’ formula for updating probabilities based on the occurrence of an event O is:

[image]

Equivalently, based on the tree diagram below, [image]

[image]

The number of ways to order n objects is n factorial, n! = n × (n − 1) × (n − 2) × ... × 1.

There are [image] ways to assign k different labels to n items, where n_i is the number of items with the label i.

The number of ways to choose a subset of size r from a set of size n when order doesn’t matter is [image] combinations; when order matters, there are [image] permutations.

A probability distribution lists all the possible outcomes of an experiment, along with their associated probabilities.

A discrete random variable has positive probabilities associated with a finite number of outcomes.

A continuous random variable has positive probabilities associated with a range of outcome values—the probability of any single value is zero.

A discrete uniform distribution is one where there are n discrete, equally likely outcomes.

The binomial distribution is a probability distribution for a binomial (discrete) random variable that has two possible outcomes.

For a discrete uniform distribution with n possible outcomes, the probability for each outcome equals 1/n.

For a binomial distribution, if the probability of success is p, the probability of x successes in n trials is:

p(x) = P(X = x) = [image]

A binomial tree illustrates the probabilities of all the possible values that a variable (such as a stock price) can take on, given the probability of an up-move and the magnitude of an up-move (the up-move factor).

[image]

With an initial stock price S = 50, U = 1.01, D = [image], and prob(U) = 0.6, the possible stock prices after two periods are:

[image]

A continuous uniform distribution is one where the probability of X occurring in a possible range is the length of the range relative to the total of all possible values. Letting a and b be the lower and upper limit of the uniform distribution, respectively, then for:

[image]

The normal probability distribution and normal curve have the following characteristics:

• The normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution.
• Mean = median = mode, and all are in the exact center of the distribution.
• The normal distribution can be completely defined by its mean and standard deviation because the skew is always zero and kurtosis is always 3.

Multivariate distributions describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable.

The correlation(s) of a multivariate distribution describes the relation between the outcomes of its variables relative to their expected values.

A confidence interval is a range within which we have a given level of confidence of finding a point estimate (e.g., the 90% confidence interval for X is X − 1.65s to X+ 1.65s).

Confidence intervals for any normally distributed random variable are:

• 90%: μ ± 1.65 standard deviations.
• 95%: μ ± 1.96 standard deviations.
• 99%: μ ± 2.58 standard deviations.

The probability that a normally distributed random variable X will be within A standard deviations of its mean, μ, [i.e., P(μ − Aσ ≤ X ≤ μ + Aσ)], is calculated as two times [1 − the cumulative left–hand tail probability, F(−A)], or two times {1 − the right-hand tail probability, [1 − F(A)]}, where F(A) is the cumulative standard normal probability of A.

The standard normal probability distribution has a mean of 0 and a standard deviation of 1.

A normally distributed random variable X can be standardized as Z =[image] and Z will be normally distributed with mean = 0 and standard deviation 1.

The z-table is used to find the probability that X will be less than or equal to a given value.

• P(X < x) = F(x) = F[image]= F(z), which is found in the standard normal probability table.
• P(X > x) = 1 – P(X < x) = 1 – F(z)

The safety-first ratio for portfolio P, based on a target return R_T, is:

[image]

Shortfall risk is the probability that a portfolio’s value (or return) will fall below a specific value over a given period of time. Greater safety-first ratios are preferred and indicate a smaller shortfall probability. Roy’s safety-first criterion states that the optimal portfolio minimizes shortfall risk.

As we decrease the length of discrete compounding periods (e.g., from quarterly to monthly) the effective annual rate increases. As the length of the compounding period in discrete compounding gets shorter and shorter, the compounding becomes continuous, where the effective annual rate = eⁱ – 1.

For a holding period return (HPR) over any period, the equivalent continuously compounded rate over the period is ln(1 + HPR).

Simple random sampling is a method of selecting a sample in such a way that each item or person in the population being studied has the same probability of being included in the sample.

A sampling distribution is the distribution of all values that a sample statistic can take on when computed from samples of identical size randomly drawn from the same population.

Time-series data consists of observations taken at specific and equally spaced points in time.

Cross-sectional data consists of observations taken at a single point in time.

Point estimates are single value estimates of population parameters. An estimator is a formula used to compute a point estimate.

Confidence intervals are ranges of values, within which the actual value of the parameter will lie with a given probability.

confidence interval = point estimate ± (reliability factor × standard error)

The reliability factor is a number that depends on the sampling distribution of the point estimate and the probability that the point estimate falls in the confidence interval.

The t-distribution is similar, but not identical, to the normal distribution in shape—it is defined by the degrees of freedom and has fatter tails compared to the normal distribution.

Degrees of freedom for the t-distribution are equal to n − 1. Student’s t–distribution is closer to the normal distribution when df is greater, and confidence intervals are narrower when df is greater.

For a normally distributed population, a confidence interval for its mean can be constructed using a z-statistic when variance is known, and a t-statistic when the variance is unknown. The z-statistic is acceptable in the case of a normal population with an unknown variance if the sample size is large (30+).

In general, we have:

• [image]when the variance is known, and
• [image] when the variance is unknown and the sample standard deviation must be used.

Increasing the sample size will generally improve parameter estimates and narrow confidence intervals. The cost of more data must be weighed against these benefits, and adding data that is not generated by the same distribution will not necessarily improve accuracy or narrow confidence intervals.

Potential mistakes in the sampling method can bias results.

These biases include data mining (significant relationships that have occurred by chance), sample selection bias (selection is non-random), look-ahead bias (basing the test at a point in time on data not available at that time), survivorship bias (using only surviving mutual funds, hedge funds, etc.), and time-period bias (the relation does not hold over other time periods).

The hypothesis testing process requires a statement of a null and an alternative hypothesis, the selection of the appropriate test statistic, specification of the significance level, a decision rule, the calculation of a sample statistic, a decision regarding the hypotheses based on the test, and a decision based on the test results.

The null hypothesis is what the researcher wants to reject.

The alternative hypothesis is what the researcher wants to prove, and it is accepted when the null hypothesis is rejected.

A two-tailed test results from a two-sided alternative hypothesis (e.g., H_a: μ ≠ μ₀). A one-tailed test results from a one-sided alternative hypothesis (e.g., H_a: μ > μ₀, or H_a: μ < μ₀).

The test statistic is the value that a decision about a hypothesis will be based on. For a test about the value of the mean of a distribution:

[image]

A Type I error is the rejection of the null hypothesis when it is actually true, while a Type II error is the failure to reject the null hypothesis when it is actually false.

The significance level can be interpreted as the probability that a test statistic will reject the null hypothesis by chance when it is actually true (i.e., the probability of a Type I error).

A significance level must be specified to select the critical values for the test.

Hypothesis testing compares a computed test statistic to a critical value at a stated level of significance, which is the decision rule for the test.

The power of a test is the probability of rejecting the null when it is false. The power of a test = 1 − P(Type II error).

A hypothesis about a population parameter is rejected when the sample statistic lies outside a confidence interval around the hypothesized value for the chosen level of significance.

The test of a hypothesis about the population variance for a normally distributed population uses a chi-square test statistic:[image], where n is the sample size, s² is the sample variance, and [image] is the hypothesized value for the population variance. Degrees of freedom are n − 1.

The test comparing two variances based on independent samples from two normally distributed populations uses an F-distributed test statistic: [image], where [image] is the variance of the first sample and [image] is the (smaller) variance of the second sample.

Underlying all of technical analysis are the following assumptions:

• Prices are determined by investor supply and demand for assets.
• Supply and demand are driven by both rational and irrational behavior.
• While the causes of changes in supply and demand are difficult to determine, the actual shifts in supply and demand can be observed in market prices.
• Prices move in trends and exhibit patterns that can be identified and tend to repeat themselves over time.

In an uptrend, prices are reaching higher highs and higher lows. An uptrend line is drawn below the prices on a chart by connecting the increasing lows with a straight line.

In a downtrend, prices are reaching lower lows and lower highs. A downtrend line is drawn above the prices on a chart by connecting the decreasing highs with a straight line.

Support and resistance are price levels or ranges at which buying or selling pressure is expected to limit price movement. Commonly identified support and resistance levels include trendlines and previous high and low prices.

The change in polarity principle is the idea that breached resistance levels become support levels and breached support levels become resistance levels.

Technical analysts look for recurring patterns in price charts.

Head-and-shoulders patterns, double tops, and triple tops are thought to be reversal patterns at the ends of uptrends. Inverse head-and-shoulders patterns, double bottoms, and triple bottoms are thought to be reversal patterns at the ends of downtrends.

Triangles, rectangles, flags, and pennants are thought to be continuation patterns, which indicate that the trend in which they appear is likely to go further in the same direction.

Price-based indicators include moving averages, Bollinger bands, and momentum oscillators such as the Relative Strength Index, moving average convergence/divergence lines, rate-of-change oscillators, and stochastic oscillators. These indicators are commonly used to identify changes in price trends, as well as “overbought” markets that are likely to decrease in the near term and “oversold” markets that are likely to increase in the near term.

Sentiment indicators include opinion polls, the put/call ratio, the volatility index, margin debt, and the short interest ratio. Margin debt, the Arms index, the mutual fund cash position, new equity issuance, and secondary offerings are flow-of-funds indicators. Technical analysts often interpret these indicators from a “contrarian” perspective, becoming bearish when investor sentiment is too positive and bullish when investor sentiment is too negative.