FRM resources

Market, credit, liquidity and operational risk

• The central limit theorem and its conditions

  Separate an asymptotic approximation from exact mean and variance identities.

  Lessons

• A unit inclusion probability in two-stage sampling

  Multiply the selected-group chance by the conditional within-group chance.

  Worked examples

• One-stage and two-stage cluster sampling

  Record group selection and within-group selection as separate stages.

  Lessons

• Sampling coverage and nonresponse

  Separate units absent from the selection frame from selected units that do not supply data.

  Concepts

• Correlated observations and effective information

  Inspect a simple common-correlation model before using a row count as independent sample size.

  Concepts

• Probability sampling design and estimation

  Connect the selection mechanism with the estimator and its uncertainty.

  Lessons

• Estimating after disproportionate stratum sampling

  Compare population-weighted and unweighted estimates when small groups are oversampled.

  Worked examples

• Enumerating a finite sampling distribution

  List every possible sample and calculate the distribution of its mean.

  Worked examples

• Defining simple random sampling only by equal chances

  Check the allowed subsets before choosing a variance formula.

  Common mistakes

• Finite-population variance of a simple random mean

  Keep the population divisor and without-replacement sample rule explicit.

  Formulas

• Systematic sampling and a periodic list

  Compare exact uncertainty under a periodic systematic design and a simple random design.

  Worked examples
• 

  Sampling plan and frame log

  Download an editable CSV for the population, selection stages and response counts.

  Templates

• Population, sampling frame and observation unit

  Define who belongs to the study and what one observation represents.

  Lessons

• Practice: sampling design and estimation

  Choose population weights, classify group selection and inspect a sampling distribution.

  Practice

• A rare-event sample mean at thirty observations

  Inspect an exact distribution that a blanket normality rule would miss.

  Worked examples

• Sampling with and without replacement

  Define whether the same population unit can be selected more than once.

  Concepts

• Random design and sample balance

  Distinguish a probability mechanism from the observed composition of one sample.

  Concepts

• Random selection and random assignment

  Distinguish choosing study units from allocating an intervention among them.

  Comparisons

• What defines a simple random sample?

  Inspect the probabilities of complete samples as well as individual-unit chances.

  Concepts

• Stratified and cluster selection

  Compare what is selected from each group before deciding which design is being described.

  Comparisons
• 

  Stratum allocation and estimation sheet

  Download a CSV that separates population shares from sampled counts.

  Templates

• Stratified sampling and population weights

  Separate subgroup sample allocation from the shares used in the population estimate.

  Lessons

• Population-weighted stratum mean

  Combine stratum sample means using defined population-unit shares.

  Formulas

• Systematic sampling: random start and list order

  Inspect the interval, start rule and ordering before treating a sample as random.

  Concepts

• Treating thirty observations as a normality guarantee

  Read the theorem conditions and actual approximation quality.

  Common mistakes

• Building a sampling-design work sample

  Show how a research question becomes a frame, selection plan and defensible estimate.

  Careers
• 

  Formula recall and conditions worksheet

  Download a sheet for variables, units, boundary checks and independent attempts.

  Templates
• 

  Mock-exam error log and review worksheet

  Download an editable CSV and turn practice errors into specific review actions.

  Templates
• 

  Operating-ratio definition log

  Download a worksheet for flow bases, averages, proxies and interpretation.

  Templates
• 

  Retake review and triage worksheet

  Download a structured record for evidence, corrections and fresh tasks.

  Templates
• 

  Exclusion-screen review worksheet

  Download an editable CSV for thresholds, missing values and screening decisions.

  Templates
• 

  Weekly study-block worksheet

  Download an editable plan that records tasks, outputs and actual time.

  Templates
• 

  Finance work-sample outline

  Download an editable outline for a question, inputs, method and limitations.

  Templates
• 

  Equal-payment annuity value calculator

  Compare end-period and beginning-period payment patterns at defined dates.

  Tools
• 

  Bayes probability update calculator

  Combine a prior with evidence probabilities under an event and its complement.

  Tools
• 

  Bond price calculator

  Price a regular fixed-rate bond and inspect its cash-flow sensitivity.

  Tools
• 

  Duration and convexity calculator

  Compare duration estimates with the exact repricing of a regular bond.

  Tools
• 

  Cash conversion cycle calculator

  Compute inventory, receivable and payable days with explicit period flows.

  Tools
• 

  DuPont analysis calculator

  Reconcile three and five factors with direct return on equity.

  Tools
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  Expected credit loss calculator

  Explore the simple PD × LGD × EAD loss model.

  Tools